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HARMONIC ANALYSIS OF ADDITIVE LEVY PROCESSES

DAVAR KHOSHNEVISAN AND YIMIN XIAO

Abstract. Let X1, . . . , XN denote N independent d-dimensional Levy processes, and con-sider the N -parameter random field

X(t) := X1(t1) + · · ·+XN (tN ).

First we demonstrate that for all nonrandom Borel sets F ⊆ Rd, the Minkowski sumX(RN

+ ) ⊕ F , of the range X(RN+ ) of X with F , can have positive d-dimensional Lebesgue

measure if and only if a certain capacity of F is positive. This improves our earlier jointeffort with Yuquan Zhong by removing a certain condition of symmetry in [68]. Moreover,we show that under mild regularity conditions, our necessary and sufficient condition canbe recast in terms of one-potential densities. This rests on developing results in classical[non-probabilistic] harmonic analysis that might be of independent interest. As was shownin [68], the potential theory of the type studied here has a large number of consequences inthe theory of Levy processes. Presently, we highlight a few new consequences.

1. Introduction

1.1. Background. It is known that, for all integers d ≥ 2, the range of d-dimensional

Brownian motion has zero Lebesgue measure. See Levy [75] for d = 2, Ville [97] for d = 3, and

Kakutani [59] for the remaining assertions. There is a perhaps better-known, but equivalent,

formulation of this theorem: When d ≥ 2, the range of d-dimensional Brownian motion does

not hit points. Kakutani [60] has generalized this by proving that, for all integers d ≥ 1, the

range of d-dimensional Brownian motion can hit a nonrandom Borel set F ⊆ Rd if and only

if cap(F ) > 0, where cap denotes, temporarily, the logarithmic capacity if d = 2 and the

Riesz capacity of index d − 2 if d ≥ 3; the case d = 1 is elementary. [Actually, Kakutani’s

paper discusses only the planar case. The theorem, for d ≥ 3, is pointed out in Dvoretzky,

Erdos, and Kakutani [22].]

Kakutani’s theorem is the starting point of a deep probabilistic potential theory initiated by

Hunt [50–53]. The literature on this topic is rich and quite large; see, for example, the books

Date: Submitted draft: June 24, 2007; final draft: July 22, 2008.1991 Mathematics Subject Classification. 60G60, 60J55, 60J45.Key words and phrases. Additive Levy processes, multiplicative Levy processes, capacity, intersections ofregenerative sets.Research supported in part by a grant from the National Science Foundation (DMS-0706728).

1

2 KHOSHNEVISAN AND XIAO

by Blumenthal and Getoor [7], Doob [10], Fukushima, Oshima, and Takeda [31], Getoor [32],

and Rockner [86], together with their combined bibliography.

One of the central assertions of probabilistic potential theory is that a nice Markov process

will hit a nonrandom measurable set F if and only if cap(F ) > 0, where cap is a certain

natural capacity in the sense of G. Choquet; see Dellacherie and Meyer [9, Chapter III, pp.

51–55]. Moreover, that capacity is defined solely, and fairly explicitly, in terms of the Markov

process itself.

There are interesting examples where F is itself random. For instance, suppose X is d-

dimensional standard Brownian motion, and F = Y ((0 ,∞)) is the range—minus the starting

point—of an independent standard Brownian motion Y on Rd. In this particular case, it is

well known that

(1.1) P X(s) = Y (t) for some s, t > 0 > 0 if and only if d ≤ 3.

This result was proved by Levy [75] for d = 2, Kakutani [59] for d ≥ 5, and Dvoretzky,

Erdos, and Kakutani [22] for d = 3, 4. Peres [83,84] and Khoshnevisan [62] contain different

elementary proofs of this fact.

There are many generalizations of (1.1) in the literature. For example, Dvoretzky, Erdos,

and Kakutani [21] proved that the paths of an arbitrary number of independent planar

Brownian motions can intersect. While Le Gall [73] proved that the trajectories of a planar

Brownian motion can intersect itself countably many times. And Dvoretzky, Erdos, Kaku-

tani, and Taylor [23] showed that three independent Brownian-motion trajectories in Rd

can intersect if and only if d ≤ 2. For other results along these lines see Hawkes [36–38],

Hendricks [39,40], Kahane [58, Chapter 16, Section 6], Lawler [69–71], Pemantle, Peres, and

Shapiro [81], Le Gall [72], Peres [82–84], Rogers [87], Tongring [96], and their combined

bibliographies.

For a long time, a good deal of effort was concentrated on generalizing (1.1) to other

concrete Markov processes than Brownian motion. But the problem of deciding when the

paths of N independent (general but nice) Markov processes can intersect remained elusive.

It was finally settled, more than three decades later, by Fitzsimmons and Salisbury [29],

whose approach was to consider the said problem as one about a certain multiparameter

Markov process.

To be concrete, let us consider the case N = 2, and let X := X(t)t≥0 and Y :=

Y (t)t≥0 denote two independent (nice) Markov processes on a (nice) state space S. The

starting point of the work of Fitzsimmons and Salisbury is the observation that PX(s) =

Y (t) for some s, t > 0 > 0 if and only if the two-parameter Markov process X ⊗ Y hits the

HARMONIC ANALYSIS OF ADDITIVE LEVY PROCESSES 3

diagonal diag S := x⊗ x : x ∈ S of S × S, where

(1.2) (X ⊗ Y )(s , t) :=

(X(s)

Y (t)

)for all s, t ≥ 0.

In the special case that X and Y are Levy processes, the Fitzsimmons–Salisbury theory was

used to solve the then-long-standing Hendricks–Taylor conjecture [41].

The said connection to multiparameter processes is of paramount importance in the

Fitzsimmons–Salisbury theory, and appears earlier in the works of Evans [24, 25]. See also

Le Gall, Shieh, and Rosen [74] and Salisbury [90–92]. Walsh [98, pp. 364–368] discusses a

connection between the random field X ⊗ Y and the Dirichlet problem for the bi-Laplacian

∆⊗∆ on the bi-disc of Rd ×Rd.

The Fitzsimmons–Salisbury theory was refined and generalized in different directions by

Hirsch [42], Hirsch and Song [43–49], and Khoshnevisan [63, 64]. See also Ren [85] who

derives an implicit-function theorem in classical Wiener space by studying a very closely-

related problem.

The two-parameter process X ⊗ Y itself was introduced earlier in the works of Wolpert

[102], who used X ⊗ Y to build a (φκ)2-model of Euclidean field theory. This too initiated

a very large body of works. For some of the earlier examples, see the works by Aizenman

[1], Albeverio and Zhou [2], Dynkin [11–20], Felder and Frohlich [28], Rosen [88, 89], and

Westwater [99–101]. [This is by no means an exhaustive list.]

In the case that X and Y are Levy processes on Rd [i.e., have stationary independent

increments], X⊗Y is an example of a so-called additive Levy process. But as it turns out, it

is important to maintain a broader perspective and consider more than two Levy processes.

With this in mind, let X1, . . . , XN denote N independent Levy processes on Rd such that

each Xj is characterized via the Levy–Khintchine formula [6, 93]:

(1.3) E exp (iξ ·Xj(t)) = exp (−tΨj(ξ)) for all t ≥ 0 and ξ ∈ Rd.

The function Ψj is called the characteristic exponent—or Levy exponent—of Xj, and its

defining property is that Ψj is a negative-definite function in the classical sense of Schoenberg

[94]; see Berg and Forst [3, Chapter II] for a pedagogic treatment.

1.2. The main results. The main object of this paper is to develop some basic probabilistic

potential theory for the following N -parameter random field X with values in Rd:

(1.4) X(t) := X1(t1) + · · ·+XN(tN) for all t := (t1 , . . . , tN) ∈ RN+ .

On a few occasions we might write (⊕Nj=1Xj)(t) in place of X(t), as well.


The random field X is a so-called additive Levy process, and is characterized by its multi-

parameter Levy–Khintchine formula:

(1.5) E exp (iξ · X(t)) = exp (−t ·Ψ(ξ)) for all t ∈ RN+ and ξ ∈ Rd;

where Ψ(ξ) := (Ψ1(ξ) , . . . ,ΨN(ξ)) is the characteristic exponent of X. Our goal is to de-

scribe the potential-theoretic properties of X solely in terms of its characteristic exponent

Ψ. Thus, it is likely that our harmonic-analytic viewpoint can be extended to study the

potential theory of more general multiparameter Markov processes that are based on the

Feller processes of Jacob [54–56].

In order to describe our main results let us first consider the kernel

(1.6) KΨ(ξ) :=N∏j=1

Re

(1

1 + Ψj(ξ)

)for all ξ ∈ Rd.

When N = 1, this kernel plays a central role in the works of Orey [80] and Kesten [61]. The

kernel for general N was introduced first by Evans [25]; see also Khoshnevisan, Xiao, and

Zhong [68].

Based on the kernel KΨ, we define, for all Schwartz distributions µ on Rd,

(1.7) IΨ(µ) :=1

(2π)d

∫Rd

|µ(ξ)|2KΨ(ξ) dξ,

provided that the Fourier transform µ of µ is a function.

We are primarily interested in the case where µ is a real-valued locally integrable function,

or a σ-finite Borel measure on Rd. In either case, we refer to IΨ(µ) as the energy of µ. Our

notion of energy corresponds to a capacity capΨ, which is the following set function: For all

Borel sets F ⊆ Rd,

(1.8) capΨ(F ) :=

[inf

µ∈Pc(F )IΨ(µ)

]−1

,

where Pc(F ) denotes the collection of all compactly-supported Borel probability measures

on F , inf ∅ :=∞, and 1/∞ := 0.

The following is the first central result of this paper. Here and throughout, λk denotes

k-dimensional Lebesgue measure on Rk for all integers k ≥ 1.

Theorem 1.1. Let X be an N-parameter additive Levy process on Rd with exponent Ψ.

Then, for all Borel sets F ⊆ Rd,

(1.9) E[λd(X(RN

+ )⊕ F)]> 0 if and only if capΨ(F ) > 0.


Remark 1.2. (1) Theorem 1.1 in the one-parameter setting is still very interesting, but

much easier to derive. See Kesten [61] for the case that F := 0 and Hawkes [33]

for general F . For a scholarly pedagogic account see the book by Bertoin [6, p. 60].

(2) One can view Theorem 1.1 as a contribution to the theory of Dirichlet forms for

a class of infinite-dimensional Levy processes. These Levy processes are in general

non-symmetric. Rockner [86] describes a general theory of Dirichlet forms for nice

infinite-dimensional Markov processes that are not necessarily symmetric. It would

be interesting to know if the processes of the present paper lend themselves to the

analysis of the general theory of Dirichlet forms. We have no conjectures along these

lines.

Our earlier collaborative effort with Yuquan Zhong [68] yielded the conclusion of Theorem

1.1 under an additional technical condition on X1, . . . , XN . A first aim of this paper is to

establish the fact that Theorem 1.1 holds in complete generality. Also, we showed in our

earlier works [66–68] that such a theorem has a large number of consequences, many of them

in the classical theory of Levy processes itself. Next we describe a few such consequences

that are nontrivial due to their intimate connections to harmonic analysis.

Our next result provides a criterion for a Borel set F ⊆ Rd to contain intersection points

of N independent Levy processes. It completes and complements the well-known results of

Fitzsimmons and Salisbury [29]. See also Corollary 9.3 and Remark 9.4 below.

Theorem 1.3. Let X1, . . . , XN be independent Levy processes on Rd, and assume that each

Xj has a one-potential density uj : Rd → R+ such that uj(0) > 0. Then, for all nonempty

Borel sets F ⊆ Rd,

(1.10) P X1(t1) = · · · = XN(tN) ∈ F for some t1, . . . , tN > 0 > 0

if and only if there exists a compact-support Borel probability measure µ on F such that

(1.11)

∫Rd

· · ·∫

Rd

∣∣µ (ξ1 + · · ·+ ξN)∣∣2 N∏

j=1

Re

(1

1 + Ψj(ξj)

)dξ1 · · · dξN <∞.

Suppose, in addition, that every uj is continuous on Rd, and finite on Rd \ 0. Then,

another equivalent condition is that there exists a compact-support probability measure µ on

F such that

(1.12)

∫∫ N∏j=1

(uj (x− y) + uj (y − x)

2

)µ(dx)µ(dy) <∞.


In order to describe our next contribution, let us recall that the one-potential measure U

of a Levy process X := X(t)t≥0 on Rd is defined as

(1.13) U(A) :=

∫ ∞0

P X(t) ∈ A e−t dt,

for all Borel sets A ⊆ Rd. Next we offer a two-parameter “additive variant” which requires

fewer technical conditions than Theorem 1.3.

Theorem 1.4. Suppose X1 and X2 are independent Levy processes on Rd with respective

one-potential measures U1 and U2. Suppose U1(dx)/dx = u1(x), where u1 : Rd → R+, and

u1 ∗ U2 > 0 almost everywhere. Then, for all Borel sets F ⊆ Rd,

(1.14) P X1(t1) +X2(t2) ∈ F for some t1, t2 > 0 > 0


(1.15)

∫Rd

|µ(ξ)|2 Re

(1

1 + Ψ1(ξ)

)Re

(1

1 + Ψ2(ξ)

)dξ <∞.

Suppose, in addition, that u1 is continuous on Rd, and finite on Rd \ 0. Then, (1.14)

holds if and only if there exists a compact-support probability measure µ on F such that

(1.16)

∫∫Q(x− y)µ(dx)µ(dy) <∞,

where

(1.17) Q(x) :=

∫Rd

[u1(x+ y) + u1(x− y) + u1(−x+ y) + u1(−x− y)

4

]U2(dy)

for all x ∈ Rd.

Among other things, Theorem 1.4 confirms a conjecture of Bertoin ([5] and [4, p. 49]); see

Remark 8.1 for details.

Finally we mention a result on the Hausdorff dimension of the set of intersections of the

sample paths of Levy processes.

Theorem 1.5. Let X1, . . . , XN be independent Levy processes on Rd, and assume that each

Xj has a one-potential density uj : Rd → R+ such that uj(0) > 0. Then, almost surely on

∩Nk=1Xk(R+) 6= ∅,

dimH

N⋂k=1

Xk(R+)

= sup

s ∈ (0 , d) :

∫(Rd)N

N∏j=1

Re

(1

1 + Ψj(ξj)

)dξ

1 + ‖ξ1 + · · ·+ ξN‖d−s<∞

,

(1.18)


where sup ∅ := 0. Suppose, in addition, that the uj’s are continuous on Rd, and finite on

Rd \ 0. Then, almost surely on ∩Nk=1Xk(R+) 6= ∅,

(1.19) dimH

N⋂k=1

Xk(R+) = sup

s ∈ (0 , d) :

∫Rd

N∏j=1

(uj(z) + uj(−z)

2

)dz

‖z‖s<∞

.

In the remainder of the paper we prove Theorem 1.1 and its stated corollaries in the order

in which they are presented. Finally, we conclude with two zero-one laws for the Lebesgue

measure and capacity of the range of an additive Levy process, that, we believe, might have

independent interest.

We end this section with four problems and conjectures.

Open problem 1. Throughout this paper, we impose continuity conditions on various one-

potential densities. This is mainly because we are able to develop general harmonic-analytic

results only for kernels that satisfy some regularity properties. Can the continuity conditions

be dropped? We believe the answer is “yes.” This is motivated, in part, by the following

fact, which follows from inspecting the proofs: The condition “u is continuous on Rd and

finite on Rd \0” is used only for proving the “if” portions in the second parts of Theorems

1.3 and 1.4.

Open problem 2. Jacob [54–56] has constructed a very large class of Feller processes

that behave locally like Levy processes. Moreover, his construction is deeply connected to

harmonic analysis. Because the results of the present paper involve mainly the local structure

of Levy processes, and are inextricably harmonic analytic, we ask: Is it possible to study

the harmonic-analytic potential theory of several Jacob processes by somehow extending the

methods of the present paper?

Open problem 3. We ask: Is there a “useful” theory of excessive functions and/or mea-

sures for additive Levy processes (or more general multiparameter Markov processes)? This

question is intimately connected to Open Problem 1, but deserves to be asked on its own.

In the one-parameter case, the answer is a decisive “yes”; confer with Getoor [32]. But the

one-dimensional theory does not appear to readily have a suitable extension to the multipa-

rameter setting.

Open problem 4. We conjecture that, under the conditions of Theorem 1.5, the following

holds almost surely on ∩Nk=1Xk(R+) 6= ∅:

(1.20) dimH

N⋂k=1

Xk(R+) = sup

s ∈ (0 , d) :

∫(−1,1)d

N∏j=1

(uj(z) + uj(−z)

2

)dz

‖z‖s<∞

.


[The difference between this and (1.19) is in the range of the integrals.] But in all but one

case we have no proof; see Remark 9.6 below for the mentioned case. As we shall see in that

remark, what we actually prove is the following harmonic-analytic fact: Suppose u is the

one-potential density of a Levy process, u(0) > 0, u is continuous on Rd, and u is finite on

Rd \ 0. Then the local square-integrability of u implies the [global] square-integrability of

u. We believe that the following more general result holds: If u1, . . . , uN are one-potential

densities that share the stated properties for u, then

(1.21)N∏j=1

(uj(•) + uj(−•)

2

)∈ L1

loc(Rd) ⇒

N∏j=1

(uj(•) + uj(−•)

2

)∈ L1(Rd).

If this is so, then the results of this paper imply Conjecture (1.20).

2. The stationary additive Levy random field

Consider a classical Levy process X := X(t)t≥0 on Rd with characteristic exponent Ψ.

Let us introduce an independent copy X ′ of X, and extend the definition of X to a process

indexed by R as follows:

(2.1) X(t) :=

X(t) if t ≥ 0,

−X ′(−t) if t < 0.

This is the two-sided Levy process with exponent Ψ in the sense that X := X(t)t∈R has

stationary and independent increments. Moreover, X(t+ s)− X(s)t≥0 is a copy of X for

all s ∈ R.

We also define X := X(t)t∈RN as the corresponding N -parameter process, indexed by

all of RN , whose values are in Rd and are defined as

(2.2) X(t) := X1(t1) + · · ·+ XN(tN) for all t := (t1 , . . . , tN) ∈ RN .

We are assuming, of course, that X1, . . . , XN are independent two-sided extensions of the

processes X1, . . . , XN , respectively.

We intend to prove the following two-sided version of Theorem 1.1.

Theorem 2.1. For all Borel sets F ⊆ Rd,

(2.3) E[λd

(X(RN)⊕ F

)]> 0 if and only if capΨ(F ) > 0.

This implies Theorem 1.1 effortlessly. Indeed, we know already from Remark 1.2 of Khosh-

nevisan, Xiao, and Zhong [68] that

(2.4) capΨ(F ) > 0 =⇒ E[λd(X(RN

+ )⊕ F)]> 0.


Thus, we seek only to derive the converse implication. But that follows from Theorem 2.1,

because X(RN+ ) ⊆ X(RN).

Henceforth, we assume that the underlying probability space Ω is the collection of all paths

ω : RN → Rd that have the form ω(t) =∑N

j=1 ωj(tj) for all t ∈ RN , where each ωj maps

RN to Rd such that ωj(0) = 0; and ωj ∈ DRd(R), the Skorohod space of cadlag functions

from R—not [0 ,∞)—to Rd. The space Ω inherits its Borel σ-algebra from the Skorohod

topology on DRd(R) in a standard way.

We can then assume that the stationary additive Levy fields, described earlier in this

section, are in canonical form. That is, X(t)(ω) := ω(t) for all t ∈ RN and ω ∈ Ω. Because

we are interested only in distributional results, this is a harmless assumption.

Define Px to be the law of x + X, and Ex the expectation operation with respect to Px,

for every x ∈ Rd. Thus, we are identifying P with P0, and E with E0.

We are interested primarily in the σ-finite measure

(2.5) Pλd :=

∫Rd

Px dx,

and the corresponding expectation operator Eλd , defined by

(2.6) Eλdf :=

∫Ω

f(ω) Pλd(dω) for all f ∈ L1(Pλd).

If AB := a− b : a ∈ A, b ∈ B, then by the Fubini–Tonelli theorem,

E[λd

(X(RN) F

)]= E

[∫Rd

1eX(RN )F (x) dx

]=

∫Rd

P−x

X(t) ∈ F for some t ∈ RN

dx

= Pλd

X(RN) ∩ F 6= ∅

.

(2.7)

Thus, Theorem 2.1 is a potential-theoretic characterization of all polar sets for X under the

σ-finite measure Pλd . With this viewpoint in mind, we proceed to introduce some of the

fundamental objects that are related to the process X.

Define, for all t ∈ RN , the linear operator Pt as follows:

(2.8) (Ptf)(x) := Ex

[f(X(t)

)]for all x ∈ Rd.

This is well defined, for example, if f : Rd → R+ is Borel-measurable, or if f : Rd → R is

Borel-measurable and Pt(|f |) is finite at x. Also define the linear operator R by

(2.9) (Rf)(x) :=1

2N

∫RN

(Ptf)(x)e−[t] dt for all x ∈ Rd,


where

(2.10) [t] := |t1|+ · · ·+ |tN |

denotes the `1-norm of t ∈ RN . [We will use this notation throughout.] The `2-norm of

t ∈ RN will be denoted by ‖t‖.Again, (Rf)(x) is well defined if f : Rd → R+ is Borel-measurable, or if f : Rd → R is

Borel-measurable and R(|f |) is finite at x.

Our next result is a basic regularity lemma for R. It should be recognized as a multipa-

rameter version of a very well-known property of Markovian semigroups and their resolvents.

Lemma 2.2. Each Pt and R are contractions on Lp(Rd), as long as 1 ≤ p ≤ ∞.

Proof. Choose and fix j ∈ 1 , . . . , N and t ∈ R, and define µj,t to be the distribution of

the random variable −Xj(t). If f : Rd → R+ is Borel-measurable, then

(2.11) Ptf = f ∗ µ1,t1 ∗ · · · ∗ µN,tN ,

where ∗ denotes convolution. This implies readily that Pt is a contraction on Lp(Rd) for all

p ∈ [1 ,∞]. On the other hand, (2.9) gives

(2.12) ‖Rf‖Lp(Rd) ≤1

2N

∫RN

‖Ptf‖Lp(Rd)e−[t] dt.

Since Pt is a contraction on Lp(Rd), the preceding is bounded above by ‖f‖Lp(Rd).

Henceforth, let “” denote the [Schwartz] Fourier transform on any and every Euclidean

space Rk. Our normalization for the Fourier transform is given by the following:

(2.13) f(ξ) :=

∫Rk

eiξ·xf(x) dx for all f ∈ L1(Rk).

Our next result identifies KΨ as the Fourier multiplier of the operator R.

Lemma 2.3. If f ∈ L1(Rd), then (Rf)(ξ) = KΨ(ξ)f(ξ) for all ξ ∈ Rd.

Proof. Recall µj,t from the proof of Lemma 2.2. Its Fourier transform is given by

(2.14) µj,t(ξ) = exp −|t|Ψj (−sgn(t)ξ) for all ξ ∈ Rd.

Equations (2.11) and (2.14), and the Plancherel theorem together imply that

(2.15) (Ptf)(ξ) = f(ξ) exp

(−

N∑j=1

|tj|Ψj (−sgn(tj)ξ)

).


Consequently,

(Rf)(ξ) =1

2Nf(ξ)

∫RN

exp

(−

N∑j=1

|tj|[1 + Ψj (−sgn(tj)ξ)

])dt

=1

2Nf(ξ)

N∏j=1

∫ ∞0

(e−t[1+Ψj(ξ)] + e−t[1+Ψj(−ξ)]

)dt.

(2.16)

A direct computation reveals that Rf = KΨf , as asserted.

The following is a functional-analytic consequence.

Corollary 2.4. The operator R maps L2(Rd) into L2(Rd), and is self-adjoint.

Proof. By Lemma 2.3, if f ∈ L1(Rd) ∩ L2(Rd) and g ∈ L2(Rd), then

(2.17)

∫Rd

(Rf)(x)g(x) dx =1

(2π)d

∫Rd

KΨ(ξ) f(ξ) g(ξ) dξ.

Thanks to Lemma 2.2, the preceding holds for all f ∈ L2(Rd). Duality then implies that R

maps L2(Rd) to itself. Moreover, since KΨ is real, R is self-adjoint.

The following lemma shows that for every t ∈ RN , the distribution of X(t) under Pλd is

λd. This is the reason why we call X a stationary additive Levy process.

Lemma 2.5. If f : Rd → R+ is Borel-measurable, then

(2.18) Eλd

[f(X(t)

)]=

∫Rd

(Ptf)(x) dx =

∫Rd

f(y) dy for all t ∈ RN .

Proof. We apply the Fubini–Tonelli theorem to find that

(2.19) Eλd

[f(X(t)

)]= E

∫Rd

f(x+ X(t)

)dx.

A change of variables [y := x + X(t)] proves that the preceding expression is equal to the

integral of f . This implies half of the lemma. Another application of the Fubini–Tonelli

theorem implies the remaining half as well.

Let us choose and fix a subset π ⊆ 1 , . . . , N and identify π with the partial order ≺π,

on RN , which is defined as follows: For all s, t ∈ RN ,

(2.20) s ≺π t iff

si ≤ ti for all i ∈ π, and

si > ti for all i 6∈ π.


The collection of all such subsets π ⊆ 1 , . . . , N forms a collective total order on RN in

the sense that

(2.21) for all s, t ∈ RN there exists π ⊆ 1 , . . . , N such that s ≺π t.

For all π ⊆ 1 , . . . , N, we define the π-history of the random field X as the collection

(2.22) Hπ(t) := σ

(X(s)

s≺πt

)for all t ∈ RN ,

where σ( · · · ) denotes the σ-algebra generated by whatever is in the parentheses. Without

loss of generality we assume that each Hπ(t) is complete with respect to Px for all x ∈ Rd;

else, we replace Hπ(t) with the said completion. Also, we assume without loss of generality

that t 7→Hπ(t) is π-right-continuous. More precisely, we assume that

(2.23) Hπ(t) =⋂

s∈RN : t≺πs

Hπ(s) for all t ∈ RN and π ⊆ 1 , . . . , N.

If not, then we replace the left-hand side by the right-hand side everywhere.

The following is an analogue of Proposition 3.2 in Khoshnevisan, Xiao, and Zhong [68] for

X. Note that conditional expectations under the σ-finite measure Pλd

are defined in exactly

the same manner as those with respect to probability measures. That is, if A is a σ-algebra

of subsets of Ω and f ∈ L1(Pλd), then the following relation defines Eλd(f |A ) uniquely up

to Pλd-evanescent sets: Eλd(hEλd [f |A ]) = Eλd(hf) for all h ∈ L∞(Ω ,A ,Pλd).

Proposition 2.6 (A Markov-random-field property). Suppose π ⊆ 1 , . . . , N and s ≺π t,both in RN . Then, for all measurable functions f : Rd → R+,

(2.24) Eλd

[f(X(t)

) ∣∣∣ Hπ(s)]

= (Pt−sf)(X(s)

)Pλd-a.s.

Proof. Choose and fix Borel measurable functions g, φ1, . . . , φm : Rd → R+, and “N -

parameter time points” s1, . . . , sm ∈ RN such that

(2.25) sj ≺π s ≺π t for all j = 1, . . . ,m.

According to the Fubini–Tonelli theorem,

Eλd

[f(X(t)

)g(X(s)

) m∏j=1

φj

(X(sj)

)]

=

∫Rd

E

[f(x+ X(t)

)g(x+ X(s)

) m∏j=1

φj

(x+ X(sj)

)]dx

=

∫Rd

E

[f(A+ y)

m∏j=1

φj(Aj + y)

]g(y) dy,

(2.26)


where A := X(t)− X(s) and Aj := X(sj)− X(s) for all j = 1, . . . ,m.

The independent-increments property of each of the Levy processes Xj implies that A is

independent of Ajmj=1. This in turn implies that

Eλd

[f(X(t)

)g(X(s)

) m∏j=1

φj

(X(sj)

)]

=

∫Rd

E [f(A+ y)] E

[m∏j=1

φj(Aj + y)

]g(y) dy.

(2.27)

After a change of variables and an appeal to the stationary-independent property of the

increments of X1, . . . , XN and X ′1, . . . , X′N , we arrive at the following:

Eλd

[f(X(t)

)g(X(s)

) m∏j=1

φj

(X(sj)

)]

=

∫Rd

E[f(X(t− s) + y

)]Ey

[m∏j=1

φj(Aj)

]g(y) dy

=

∫Rd

(Pt−sf)(y)Ey

[m∏j=1

φj(Aj)

]g(y) dy.

(2.28)

By the monotone class theorem, we can verify that for all nonnegative Hπ(s)-measurable

random variables Y , there exists a measurable function ψ : Rd → R+, depending on Y only,

such that

(2.29) Eλd

[f(X(t)

)g(X(s)

)Y]

=

∫Rd

(Pt−sf)(y)ψ(y) g(y) dy,

for all measurable functions f, g : Rd → R+. This proves the proposition.

Lemma 2.7. If f, g ∈ L2(Rd) and t, s ∈ RN , then

(2.30) Eλd

[f(X(t)

)g(X(s)

)]=

∫Rd

(Pt−sf)(y) g(y) dy.

Proof. We may consider, without loss of generality, measurable and nonnegative functions

f, g ∈ L2(Rd). Let π denote the collection of all i ∈ 1 , . . . , N such that si ≤ ti. Then

s ≺π t, and Proposition 2.6 implies that Pλd-a.s.,

(2.31) Eλd

[f(X(t)

) ∣∣∣ Hπ(s)]

= (Pt−sf)(X(s)

).

This and Lemma 2.5 together conclude the proof.


3. The sojourn operator

Recall (2.10), and consider the “sojourn operator,”

(3.1) Sf :=1

2N

∫RN

f(X(t)

)e−[t] dt.

Our first lemma records the fact that S maps density functions to mean-one random

variables [Pλd ].

Lemma 3.1. If f is a probability density function on Rd, then Eλd [Sf ] = 1.

This follows readily from Lemma 2.5. Our next result shows that, under a mild condi-

tion on Ψj, S embeds functions in L2(Rd), quasi-isometrically, into the subcollection of all

functions in L2(Rd) that have finite energy. Namely,

Proposition 3.2. If f ∈ L2(Rd), then

(3.2) ‖Sf‖L2(Pλd ) ≤√IΨ(f).

Suppose, in addition, that there exists a constant c ∈ (1 ,√

2), such that the following sector

condition holds for all j = 1, . . . , N :

(3.3) |Im Ψj(ξ)| ≤ c (1 + Re Ψj(ξ)) for all ξ ∈ Rd.

Then, there exists a constant A ∈ (0 , 1) such that

(3.4) A√IΨ(f) ≤ ‖Sf‖L2(Pλd ) ≤

√IΨ(f).

Remark 3.3 (Generalized Sobolev spaces). When N = 1 and the Levy process in question is

symmetric, the following problem arises in the theory of Dirichlet forms: For what f in the

class D(Rd), of Schwartz distributions on Rd, can we define Sf as an element of L2(Pλd)

(say)? This problem continues to make sense in the more general context of additive Levy

processes. And the answer is given by (3.4) in Proposition 3.2 as follows: Assume that

the sector condition (3.3) holds for all j = 1, . . . , N and some c ∈ (0 ,√

2). Let SΨ(Rd)

denote the completion of the collection of all members of L2(Rd) that have finite energy

IΨ, where the completion is made in the norm ‖f‖Ψ := I1/2Ψ (f) + ‖f‖L2(Rd). Then, there

exists an a.s.-unique maximal extension S of S such that S : SΨ(Rd) → S(SΨ(Rd)) is a

quasi-isometry. The space SΨ(Rd) generalizes further some of the ψ-Bessel potential spaces

of Farkas, Jacob, and Schilling [26] and Farkas and Leopold [27]; see also Jacob and Schilling

[57], Masja and Nagel [78], and Slobodeckii [95].

The proof of Proposition 3.2 requires a technical lemma, which we develop first.


Lemma 3.4. For all z ∈ C define

(3.5) Λ(z) :=

∫ ∞−∞

∫ ∞−∞

e−|t|−|s|−|t−s|σ(z;t−s) dt ds,

where σ(z ; r) := z if r ≥ 0 and σ(z ; r) := z otherwise. Then for all z ∈ C with Re z ≥ 0,

(3.6) Λ(z) ≤ 4 Re

(1

1 + z

).

If, in addition, | Im z| ≤ c(1 + Re z) for some c ∈ (1 ,√

2), then

(3.7) Λ(z) ≥ 2(2− c2

)Re

(1

1 + z

).

Proof. The double integral is computed by dividing the region of integration into four natural

parts: (i) s, t ≥ 0; (ii) s, t ≤ 0; (iii) t ≥ 0 ≥ s; and (iv) s ≥ 0 ≥ t. Direct computation

reveals that for all z ∈ C with Re z ≥ 0∫ ∞0

∫ ∞0

e−|t|−|s|−|t−s|σ(z;t−s) dt ds+

∫ 0

−∞

∫ 0

−∞e−|t|−|s|−|t−s|σ(z;t−s) dt ds

= 2 Re

(1

1 + z

).

(3.8)

Similarly, one can compute∫ ∞0

∫ 0

−∞e−|t|−|s|−|t−s|σ(z;t−s) dt ds+

∫ 0

−∞

∫ ∞0

e−|t|−|s|−|t−s|σ(z;t−s) dt ds

=1

(1 + z)2+

1

(1 + z)2.

(3.9)

Consequently,

(3.10) Λ(z) = 2 Re

(1

1 + z

)+

2((1 + Re z)2 − (Im z)2

)|1 + z|4

,

for all z ∈ C with Re z ≥ 0. It follows that

(3.11) Λ(z) ≤ 2 Re

(1

1 + z

)[1 + Re

(1

1 + z

)]for all z ∈ C with Re z ≥ 0. Whenever Re z ≥ 0, we have 0 ≤ Re(1 + z)−1 ≤ 1, and hence

(3.6) follows from (3.11). On the other hand, if | Im z| ≤ c(1 + Re z), then (3.10) yields

(3.12) Λ(z) ≥ 2 Re

(1

1 + z

)+ 2

(1− c2

) [Re

(1

1 + z

)]2

,

from which the result follows readily, because 0 ≤ Re[(1 + z)−1] ≤ 1 when Re z ≥ 0.


Proof of Proposition 3.2. We apply Lemma 2.7 to deduce that

(3.13) Eλd

(|Sf |2

)=

1

4N

∫Rd

f(y) dy

∫RN

e−[t] dt

∫RN

e−[s] ds (Pt−sf)(y).

In accord with (2.15) and Parseval’s identity, for all u ∈ RN ,∫Rd

f(y)(Puf)(y) dy =1

(2π)d

∫Rd

f(ξ) (Puf)(ξ) dξ

=1

(2π)d

∫Rd

∣∣∣f(ξ)∣∣∣2 exp

(−

N∑j=1

|uj|Ψj (−sgn(uj)ξ)

)dξ.

(3.14)

This and the Fubini–Tonelli theorem together reveal that

(3.15) Eλd

(|Sf |2

)=

1

4N(2π)d

∫Rd

∣∣∣f(ξ)∣∣∣2 N∏j=1

Λ(Ψj(ξ)) dξ.

Since Re Ψj(ξ) ≥ 0 for all ξ ∈ Rd and j = 1, . . . , N , we apply Lemma 3.4 to this formula,

and conclude the proof of the proposition.

4. Proof of Theorem 2.1

Thanks to the definition of capΨ, and to the countable additivity of P, it suffices to

consider only the case that

(4.1) F is a compact set.

This condition is tacitly assumed throughout this section. We note, in particular, that Pc(F )

denotes merely the collection of all Borel probability measures that are supported on F .

Proposition 5.7 of Khoshnevisan, Xiao, and Zhong [68] proves that for every compact set

F ⊆ Rd,

(4.2) capΨ(F ) > 0 =⇒ E[λd(X([0, r]N)⊕ F

)]> 0

for all r > 0. It is clear that the latter property implies that E[λd(X(RN) ⊕ F )] > 0, and

one obtains half of the theorem.

Since capΨ(−F ) = capΨ(F ), we may and will replace F by −F throughout. In light of

(2.7), we then assume that

(4.3) Pλd

X(RN) ∩ F 6= ∅

> 0,

and seek to deduce the existence of a probability measure µ on F such that IΨ(µ) <∞.

We will prove a little more. Namely, that for all sufficiently large integers k > 0,

(4.4) Pλd

X([−k , k]N

)∩ F 6= ∅

≤ e2Nk16N capΨ(F ).


This will conclude our proof of the second half of the theorem.

It is a standard fact that there exists a probability density function φ1 in C∞(Rd) with

the following properties:

P1. φ1(x) = 0 if ‖x‖ > 1;

P2. φ1(x) = φ1(−x) for all x ∈ Rd;

P3. φ1(x) ≥ 0 for all x ∈ Rd.

This can be obtained, for example, readily from Plancherel’s [duality] theorem of Fourier

analysis.

We recall also the following standard fact: φ1 ∈ L1(Rd) and P3 together imply that

φ1 ∈ L1(Rd); see Hawkes [33, Lemma 1].

Now we define an approximation to the identity φεε>0 by setting

(4.5) φε(x) :=1

εdφ1

(xε

)for all x ∈ Rd and ε > 0.

It follows readily from this that for every µ ∈Pc(F ):

(i) µ ∗ φε is a uniformly continuous probability density;

(ii) µ ∗ φε is supported on the closed ε-enlargement of F , which we denote by Fε, for all

ε > 0;

(iii) φε is symmetric; and

(iv) limε→0+ φε(ξ) = 1 for all ξ ∈ Rd.

As was done in Khoshnevisan, Xiao, and Zhong [68], we can find a random variable T

with values in RN ∪ ∞ such that:

(1) T =∞ is equal to the event that X(t) 6∈ F for all t ∈ RN ;

(2) X(T ) ∈ F on T 6=∞.

This can be accomplished pathwise. Consequently, (4.3) is equivalent to the condition

that

(4.6) Pλd T 6=∞ > 0.

Define for all Borel sets A ⊆ Rd and all integers k, l ≥ 1,

(4.7) µk,l(A) :=Pλd

X(T ) ∈ A , T ∈ [−k , k]N ,

∣∣∣X(0)∣∣∣ ≤ l

Pλd

T ∈ [−k , k]N ,

∣∣∣X(0)∣∣∣ ≤ l

.

Because the Pλd-distribution of X(0) is λd, µk,l is a probability measure on F for all k and l

sufficiently large; see (4.6).


Choose and fix k and l so large that µk,l ∈ Pc(F ), and define fε := µk,l ∗ φε. According

to Proposition 2.6, for all nonrandom times τ ∈ RN ,∑π⊆1,...,N

Eλd

[∫tπτ

fε

(X(t)

)e−[t] dt

∣∣∣∣ Hπ(τ )

]

=∑

π⊆1,...,N

∫tπτ

(Pt−τfε)(X(τ )

)e−[t] dt

≥ e−[τ ]∑

π⊆1,...,N

∫sπ0

(Psfε)(X(τ )

)e−[s] ds.

(4.8)

It follows from (4.8) that for all nonrandom times τ ∈ RN ,∑π⊆1,...,N

Eλd [Sfε | Hπ(τ )] ≥ e−[τ ]

2N

∫RN

(Psfε)(X(τ )

)e−[s] ds

= e−[τ ] (Rfε)(X(τ )

)Pλd-a.s..

(4.9)

Because fε is continuous and compactly supported, one can verify from (2.9) that Rfε is

continuous [this can also be shown by Lemma 2.3, the fact that fε ∈ L1(Rd) and the Fourier

inversion formula]. Also, the L2(Pλd)-norm of the left-most term in (4.9) is bounded above

by ‖Sfε‖L2(Pλd ), and this is at most√IΨ(fε), in turn; see Proposition 3.2. It is easy to see

that

IΨ(fε) = IΨ(µk,l ∗ φε)

≤ 1

(2π)2d

∫Rd

|φε(ξ)|2 dξ

<∞.

(4.10)

Since |µk,l(ξ)|2KΨ(ξ) ≤ 1, we conclude that ‖Sfε‖L2(Pλd ) is finite for all ε > 0.

A simple adaptation of the proof of Proposition 2.6 proves that Hπ is a commuting filtration

for all π ⊆ 1 , . . . , N. By this we mean that for all t ∈ R,

(4.11) H 1π (t1), . . . ,H N

π (tN) are conditionally independent [Pλd ], given Hπ(t),

where H jπ (tj) is defined as the following σ-algebra:

(4.12) H jπ (tj) :=

σ(Xj(s); −∞ < s ≤ tj

)if j ∈ π,

σ(Xj(s); ∞ > s ≥ tj

)if j 6∈ π.

Here, σ( · · · ) denotes the σ-algebra generated by the random variables in the parenthesis.

The stated commutation property readily implies that for all random variables Y ∈L2(Pλd) and partial orders π ⊆ 1 , . . . , N:


(1) τ 7→ Eλd [Y |Hπ(τ )] has a version that is cadlag in each of its N variables, uniformly

in all other N − 1 variables; and

(2) The second moment of supτ∈RN Eλd [Y |Hπ(τ )] is at most 4N times the second mo-

ment of Y [Pλd ].

See Khoshnevisan [64, Theorem 2.3.2, p. 235] for the case where Pλd is replaced by a prob-

ability measure. The details of the remaining changes are explained in a slightly different

setting in Lemma 4.2 of Khoshnevisan, Xiao, and Zhong [68]. In summary, (4.9) holds for all

τ ∈ RN , Pλd-almost surely [note the order of the quantifiers]. It follows immediately from

this that for all integers k ≥ 1,

(4.13)∑

π⊆1,...,N

supτ∈RN

Eλd [Sfε | Hπ(τ )] ≥ e−Nk (Rfε)(X(T )

)· 1T∈[−k,k]N , |eX(0)|≤l,

Pλd-almost surely. According to Item (2) above, the second moment of each summand in

the left-hand side is at most 4N times the second moment of Sfε. As we noticed earlier, the

latter is at most IΨ(fε). Therefore, by squaring both sides of (4.13) we derive

e2Nk16NIΨ(fε)

≥ Eλd

[∣∣∣(Rfε)(X(T ))∣∣∣2 ; T ∈ [−k , k]N ,

∣∣∣X(0)∣∣∣ ≤ l

]= Eλd

[∣∣∣(Rfε)(X(T ))∣∣∣2 ∣∣∣∣ T ∈ [−k , k]N ,

∣∣∣X(0)∣∣∣ ≤ l

]× Pλd

T ∈ [−k , k]N ,

∣∣∣X(0)∣∣∣ ≤ l

.

(4.14)

It follows from this and the Cauchy–Schwarz inequality that

e2Nk16NIΨ(fε)

≥∣∣∣Eλd

[(Rfε)

(X(T )

) ∣∣∣ T ∈ [−k , k]N ,∣∣∣X(0)

∣∣∣ ≤ l]∣∣∣2

× Pλd

T ∈ [−k , k]N ,

∣∣∣X(0)∣∣∣ ≤ l

.

(4.15)

Using the definition of µk,l, we can write the above as

e2Nk16NIΨ(fε)

≥∣∣∣∣∫

Rd

(Rfε) dµk,l

∣∣∣∣2 · Pλd

T ∈ [−k , k]N ,

∣∣∣X(0)∣∣∣ ≤ l

=

1

(2π)2d

∣∣∣∣∫Rd

µk,l(ξ) (Rfε)(ξ) dξ

∣∣∣∣2 · Pλd

T ∈ [−k , k]N ,

∣∣∣X(0)∣∣∣ ≤ l

,

(4.16)

where the equality follows from the Parseval identity. According to Lemma 2.3, the Fourier

transform of Rfε is KΨ times the Fourier transform of fε, and the latter is µk,lφε. Hence


(4.16) implies that

e2Nk16NIΨ(fε)

≥ 1

(2π)2d

∣∣∣∣∫Rd

|µk,l(ξ)|2 φε(ξ) KΨ(ξ) dξ

∣∣∣∣2 · Pλd

T ∈ [−k , k]N ,

∣∣∣X(0)∣∣∣ ≤ l

.

(4.17)

Now we apply Property P3 to deduce that φε ≥ 0. Because φε is also a probability density,

it follows that φε ≥ |φε|2. Consequently,

e2Nk16NIΨ(fε)

≥ 1

(2π)2d

∣∣∣∣∫Rd

∣∣∣fε(ξ)∣∣∣2 KΨ(ξ) dξ

∣∣∣∣2 · Pλd

T ∈ [−k , k]N ,

∣∣∣X(0)∣∣∣ ≤ l

= |IΨ(fε)|2 · Pλd

T ∈ [−k , k]N ,

∣∣∣X(0)∣∣∣ ≤ l

.

(4.18)

We have seen in (4.10) that IΨ(fε) is finite for each ε > 0. If it were zero for arbitrary small

ε > 0, then we could apply Fatou’s lemma to deduce that IΨ(µk,l) ≤ lim infε→0 IΨ(fε) = 0.

This and the fact that KΨ(ξ) > 0 for all ξ ∈ Rd would imply that µk,l(ξ) ≡ 0 for all ξ ∈ Rd,

which is a contradiction. Hence we can deduce that for all ε > 0 small enough,

(4.19)e2Nk16N

IΨ(fε)≥ Pλd

T ∈ [−k , k]N ,

∣∣∣X(0)∣∣∣ ≤ l

.

Thanks to (4.6) and the monotone convergence theorem, the right-hand side is positive,

provided that we choose [and fix] sufficiently-large integers k and l.

The right-hand side of (4.19) is independent of ε > 0. Hence, we can let ε ↓ 0 and appeal

to Fatou’s lemma to deduce that IΨ(µk,l) < ∞. Thus, in any event, we have produced a

probability measure µk,l on F whose energy IΨ(µk,l) is finite. This concludes the proof. We

complete this discussion by verifying (4.4).

Equation (4.19) and the defining properties of the function T together imply that for all

sufficiently-large integers k, l ≥ 1,

(4.20) Pλd

X([−k , k]N

)∩ F 6= ∅ ,

∣∣∣X(0)∣∣∣ ≤ l

≤ e2Nk16N capΨ(F ).

Let l ↑ ∞ and appeal to the monotone convergence theorem to deduce (4.4).

5. On kernels of positive type

In this section we study kernels of positive type; they are recalled next. Here and through-

out, R+ := [0 ,∞] is defined to be the usual one-point compactification of R+ := [0 ,∞),

and is endowed with the corresponding Borel σ-algebra.


Definition 5.1. A kernel [on Rd] is a Borel measurable function κ : Rd → R+ such that

κ ∈ L1loc(R

d). We say that κ is a kernel of positive type if κ is a kernel whose Fourier

transform κ is a function that is nonnegative almost everywhere.

Clearly, every kernel κ can be redefined on a Lebesgue-null set so that the resulting

modification κ maps Rd into R+. However, we might lose some of the nice properties of κ

by doing this. A notable property is that κ might be continuous; that is, κ(x) converges—in

R+—to κ(y) as x converges to y in Rd. In this case, κ might not be continuous. From this

perspective, it is sometimes advantageous to work directly with the R+-valued function κ.

For examples we have in mind the Riesz kernels ; they are defined as follows: Choose and fix

some number α ∈ (0 , d), and then let the Riesz kernel κα of index α be

(5.1) κα(x) :=

‖x‖α−d if x 6= 0,

∞ if x = 0.

It is easy to check that κα is a continuous kernel for each α ∈ (0 , d). In fact, every κα is a

kernel of positive type, as can be seen via the following standard fact:

(5.2) κα(ξ) = cd,ακd−α(ξ) for all ξ ∈ Rd.

Here cd,α is a universal constant that depends only on d and α; see Kahane [58, p. 134] or

Mattila [79, eq. (12.10), p. 161], for example.

It is true—but still harder to prove—that for all Borel probability measures µ on Rd,

(5.3)

∫∫µ(dx)µ(dy)

‖x− y‖d−α=

cd,α(2π)d

∫Rd

|µ(ξ)|2 dξ

‖ξ‖α.

See Mattila [79, p. Lemma 12.12, p. 162].

The utility of (5.3) is in the fact that it shows that Riesz-type energies of the left-hand side

are equal to Polya–Szego energies of the right-hand side. This is a probabilistically significant

fact. For example, consider the case that α ∈ (0 , 2]. Then, µ 7→∫∫‖x− y‖−d+α µ(dx)µ(dy)

is the “energy functional” associated to continuous additive functionals of various stable

processes of index α. At the same time, µ 7→ (2π)−d∫

Rd |µ(ξ)|2 ‖ξ‖−α dξ is a Fourier-analytic

energy form of the type that appears more generally in the earlier parts of the present paper.

Roughly speaking, (2π)−d∫

Rd |µ(ξ)|2 ‖ξ‖−α dξ IΨ(µ), where Ψ(ξ) = ‖ξ‖α defines the Levy

exponent of an isotropic stable process of index α. [Analytically speaking, this is the Sobolev

norm of µ that corresponds to the fractional Laplacian operator −(−∆)α/2.] Thus, we seek

to find a useful generalization of (5.3) that goes beyond one-parameter stable processes.


We define for all finite Borel measures µ and ν on Rd, and all kernels κ on Rd,

(5.4) Eκ(µ , ν) :=

∫∫ (κ(x− y) + κ(y − x)

2

)µ(dx) ν(dy).

This is called the mutual energy between µ and ν in gauge κ, and defines a quadratic form

with pseudo-norm

(5.5) Eκ(µ) := Eκ(µ , µ).

This is the “κ-energy” of the measure µ. There is a corresponding capacity defined as

(5.6) Cκ(F ) :=1

inf Eκ(µ),

where the infimum is taken over all compactly supported probability measures µ on F ,

inf ∅ :=∞, and 1/∞ := 0.

We can recognize the left-hand side of (5.3) to be Eκd−α(µ). Because κd−α(x) <∞ if and

only if x 6= 0, the following is a nontrivial generalization of (5.3).

Theorem 5.2. Suppose κ is a continuous kernel of positive type on Rd which satisfies one

of the following two conditions:

(1) κ(x) <∞ if and only if x 6= 0;

(2) κ ∈ L∞(Rd), and κ(x) <∞ when x 6= 0.

Then for all Borel probability measures µ and ν on Rd,

(5.7) Eκ∗ν(µ) =1

(2π)d

∫Rd

κ(ξ) Re ν(ξ) |µ(ξ)|2 dξ.

Remark 5.3. If κ ∈ L1(Rd), then κ can be defined by the usual Fourier transform, κ(ξ) =∫Rd exp(ix · ξ)κ(x) dx. That is, the condition κ ∈ L∞(Rd) is verified automatically in this

case. In fact, κ is bounded in this case, as can be seen from supξ∈Rd |κ(ξ)| = ‖κ‖L1(Rd).

Our proof of Theorem 5.2 proceeds in four steps; the first three are stated as lemmas.

The folklore of harmonic analysis contains precise versions of the loose assertion that

“typically, kernels of positive type achieve their supremum at the origin.” The first step in

the proof of Theorem 5.2 is to verify a suitable form of this statement.

Lemma 5.4. Suppose κ is a kernel of positive type such that κ ∈ L∞(Rd). Suppose also

that κ is continuous on all of Rd, and finite on Rd \ 0. Then, κ(0) = supx∈Rd κ(x).

Proof. Recall the functions φε from the proof of Theorem 2.1; see (4.5). Then, κ ∗ φε ∈L1loc(R

d) and κ ∗ φε = κφε ≥ 0. But ess supξ∈Rd |κ(ξ)| <∞ and supξ∈Rd |φε(ξ)| ≤ ‖φε‖L1(Rd) =


1. Moreover, the construction of φε ensures that φε is integrable. Therefore, κ ∗ φε ∈L1(Rd) ∩ L∞(Rd), and for Lebesgue-almost all x ∈ Rd,

(5.8) (κ ∗ φε)(x) =1

(2π)d

∫Rd

e−ix·ξ κ(ξ) φε(ξ) dξ,

thanks to the inversion formula for Fourier transforms. Since both sides of (5.8) are con-

tinuous functions of x, that equation is valid for all x ∈ Rd. Furthermore, because κ is a

kernel of positive type, it follows that (κ ∗ φε)(x) ≤ (2π)−d∫

Rd κ(ξ) dξ. But if x 6= 0, then

limε↓0(κ ∗ φε)(x) = κ(x), and hence,

(5.9) supx∈Rd\0

κ(x) ≤ 1

(2π)d

∫Rd

κ(ξ) dξ.

It suffices to prove that

(5.10)1

(2π)d

∫Rd

κ(ξ) dξ ≤ κ(0).

This holds trivially if κ(0) is infinite. Therefore, we may assume without loss of generality

that κ(0) <∞. The continuity of κ ensures that it is uniformly continuous in a neighborhood

of the origin, thence we have limε↓0(κ ∗ φε)(0) = κ(0) by the classical Fejer theorem. Also,

we recall that limε↓0 φε(ξ) = 1 for all ξ ∈ Rd. We use these facts in conjunction with (5.8)

and Fatou’s lemma to deduce (5.10), and hence the lemma.

Next we present the second step in the proof of Theorem 5.2. This is another folklore fact

from harmonic analysis.

We say that a kernel κ is lower semicontinuous if there exists a sequence of continuous

functions κ1, κ2, . . . : Rd → R+ such that κn(x) ≤ κn+1(x) for all n ≥ 1 and x ∈ Rd,

such that κn(x) ↑ κ(x) for all x ∈ Rd, as n ↑ ∞. Because κn(x) is assumed to be in R+

[and not R+], our definition of lower semicontinuity is slightly different from the usual one.

Nonetheless, the following is a consequence of Lemma 12.11 of Mattila [79, p. 161].

Lemma 5.5. Suppose κ is a lower semicontinuous kernel of positive type on Rd. Then, for

all Borel probability measures µ on Rd,

(5.11) Eκ(µ) ≤ 1

(2π)d

∫Rd

κ(ξ) |µ(ξ)|2 dξ.

Our next lemma constitutes the third step of our proof of Theorem 5.2.

Lemma 5.6. Suppose κ is a continuous kernel of positive type on Rd, which satisfies one of

the following two conditions:

(1) κ(x) <∞ if and only if x 6= 0;


(2) κ ∈ L∞(Rd), and κ(x) <∞ when x 6= 0.

Then, for all Borel probability measures µ on Rd,

(5.12) Eκ(µ) =1

(2π)d

∫Rd

κ(ξ) |µ(ξ)|2 dξ.

A weaker version of this result is stated in Kahane [58, p. 134] without proof.

According to Kahane (loc. cit.), functions κ that satisfy the conditions of Lemma 5.6 are

called potential kernels. They can be defined, in equivalent terms, as kernels of positive types

that are continuous on Rd \ 0 and limx→0 κ(x) = ∞. We will not use this terminology:

the term “potential kernel” is reserved for another object.

Proof of Lemma 5.6. Regardless of whether κ satisfies condition (1) or (2), it is lower semi-

continuous. Therefore, in light of Lemma 5.5, it suffices to prove that

(5.13) Eκ(µ) ≥ 1

(2π)d

∫Rd

κ(ξ) |µ(ξ)|2 dξ.

From here on, our proof considers two separate cases:

Case 1. First, let us suppose κ satisfies condition (1) of the lemma. We strive to establish

(5.13). Without loss of generality, we may assume that

(5.14) Eκ(µ) <∞;

else there is nothing to prove. Since limx→0 κ(x) =∞, (5.14) implies that µ does not charge

singletons.

According to Lusin’s theorem, for every η ∈ (0 , 1) we can find a compact set Kη ⊆ Rd\0with µ(Kc

η) ≤ η such that κ ∗ µ is continuous—and hence uniformly continuous—on Kη.

Define for all Borel sets A ⊆ Rd,

(5.15) µη(A) =µ(A ∩Kη)

1− η.

Then µη is supported by the compact set Kη, and

1

1− η≥ µη(Kη)

=1

1− ηµ(Kη)

≥ 1.

(5.16)


Now limr→0 µη(B(x , r)) = 0 for all x ∈ Rd, where B(x , r) denotes the `2-ball of radius

r > 0 about x ∈ Rd. Therefore, a compactness argument reveals that for all η ∈ (0 , 1),

(5.17) limr↓0

supx∈Kη

µη(B(x , r)) = 0.

For otherwise we can find δ > 0 and xr ∈ Kη such that for all r > 0, µη(B(xr , r)) ≥ δ. By

compactness we can extract a subsequence r′ → 0 and x ∈ Kη such that xr′ → x. It follows

easily then µη(B(x , ε)) ≥ δ for all ε > 0, whence µη(x) ≥ δ, which contradicts the fact

that µη does not charge singletons.

Next we choose and fix y ∈ Rd and η > 0. We claim that

(5.18) supx∈Kη

∫B(x,ε)

κ(y − z)µη(dz)→ 0 as ε ↓ 0.

Indeed, if y ∈ Kη, then by the fact that (κ ∗µη)(y) <∞ and (5.17), we see that for all ρ > 0

there exists ε0 > 0 such that for all 0 < ε < ε0

(5.19) supx∈Kη

∫B(x,ε)

κ(y − z)µη(dz) ≤ ρ.

If y ∈ Kcη, then by the continuity of κ on Rd\0 and (5.17), we have

(5.20) limε↓0

supx∈Kη

∫B(x,ε)

κ(y − z)µη(dz) = 0.

By combining (5.19) and (5.20), we find that for all y ∈ Rd,

(5.21) limε↓0

supx∈Kη

∫B(x,ε)

κ(y − z)µη(dz) ≤ ρ.

Thus (5.18) follows from (5.21), because we can choose ρ as small as we want. By (5.18)

and another appeal to compactness, we obtain

(5.22) supy∈O

supx∈Kη

∫B(x,ε)

κ(y − z)µη(dz)→ 0 as ε ↓ 0,

where O ⊆ Rd is any bounded open set containing Kη. Consequently, for all k ≥ 1, we can

find εk → 0 such that

(5.23) supy∈O

supx∈Kη

∣∣∣∣(κ ∗ µη)(y)−∫B(x,εk)c

κ(y − z)µη(dz)

∣∣∣∣ ≤ 1

k.

Our second claim is that the function y 7→ κ∗µη(y) is continuous on Rd. In fact, it can be

verified directly that κ ∗ µη is continuous at every y ∈ Kcη. Now let y ∈ Kη and let yn∞n=1

be an arbitrary sequence in O such that limn→∞ yn = y. Because z 7→ κ(y − z) is uniformly


continuous on B(y , εk)c ∩Kη, we have

(5.24) limn→∞

∫B(y,εk)c

κ(yn − z)µη(dz) =

∫B(y,εk)c

κ(y − z)µη(dz).

This and (5.23) together imply that for all k ≥ 1,

(5.25) limn→∞

∣∣(κ ∗ µη)(yn)− (κ ∗ µη)(y)∣∣ ≤ 2

k.

Let k ↑ ∞ to deduce that κ ∗ µη is continuous at y ∈ Kη. Hence, we have shown that κ ∗ µηis continuous on Rd.

If 0 < ε, η < 1, then we can appeal to the Fubini–Tonelli theorem, a few times in succession,

to deduce that

(5.26) Eκ∗ψε(µη) =

∫κ(−z) (aε,η ∗ bε,η) (z) dz,

where aε,η := φε ∗ µη, bε,η := φε ∗ µη, and µη is the Borel probability measure on Rd that is

defined by

(5.27) µη(A) := µη(−A) for all A ⊆ Rd.

We observe the following elementary facts:

(1) Both aε,η and bε,η are infinitely differentiable functions of compact support;

(2) because φε is of positive type, the Fourier transform of aε,η ∗ bε,η is |φε|2|µη|2;

(3) the Fourier transform of z 7→ κ(−z) is the same as that of κ because κ is a kernel of

positive type.

Therefore, we can combine the preceding with the Parseval identity, and deduce that

Eκ∗ψε(µη) =1

(2π)d

∫Rd

κ(ξ) aε,η(ξ) bε,η(ξ) dξ

=1

(2π)d

∫Rd

κ(ξ) |φε(ξ)|2 |µη(ξ)|2 dξ.

(5.28)

We can apply the Fubini–Tonelli theorem to write the left-most term in another way, as well.

Namely,

(5.29) Eκ∗ψε(µη) =

∫(κ ∗ ψε ∗ µη) dµη.

The continuity of κ ∗ µη, and Fejer’s theorem, together imply that κ ∗ µη ∗ ψε converges to

κ ∗ µη uniformly on Kη as ε ↓ 0, and hence Eκ∗ψε(µη) converges to Eκ(µη) as ε ↓ 0. This and


(5.28) together imply that

Eκ(µη) =1

(2π)dlimε→0

∫Rd

κ(ξ) |φε(ξ)|2 |µη(ξ)|2 dξ

≥ 1

(2π)d

∫Rd

κ(ξ) |µη(ξ)|2 dξ,

(5.30)

owing to Fatou’s lemma. Since µη is (1− η)−1 times a restriction of µ, it follows that

(5.31)1

(1− η)2Eκ(µ) ≥ 1

(2π)d

∫Rd

κ(ξ) |µη(ξ)|2 dξ.

Because for all ξ ∈ Rd, ∣∣∣∣µη(ξ)− µ(ξ)

1− η

∣∣∣∣ ≤ 1

1− ηµ(Kc

η)

≤ η

1− η,

(5.32)

we deduce that limη→0 µη = µ pointwise. An appeal to Fatou’s lemma justifies (5.13), and

this completes our proof in the first case.

Case 2. Next, we consider the case that κ satisfies condition (2) of the lemma. If κ(0) =∞,

then condition (1) is satisfied, and therefore the proof is complete. Thus, we may assume

that κ(0) <∞. According to Lemma 5.4, κ is a bounded and continuous function from Rd

into R+. For all Borel sets A ⊆ Rd define

(5.33) µn(A) :=µn(A ∩ [−n , n]d

)χn

, where χn := µ([−n , n]d

).

If n > 0 is sufficiently large, then µn is a well-defined Borel probability measure on [−n , n]d,

and since κ is uniformly continuous on [−n , n]d,

1

χ2n

Eκ(µ) ≥ Eκ(µn)

= limε↓0

Eκ∗ψε(µn)

=1

(2π)dlimε↓0

∫Rd

κ(ξ) ψε(ξ) |µn(ξ)|2 dξ.

(5.34)

The first equality holds because κ ∗ ψε converges uniformly to κ on [−n , n]d, as ε ↓ 0. The

second is a consequence of the Parseval identity. Thanks to Fatou’s lemma, we have proved

that for all n > 0 sufficiently large,

(5.35)1

χ2n

Eκ(µ) ≥ 1

(2π)d

∫Rd

κ(ξ) |µn(ξ)|2 dξ.


As n tends to infinity, χn ↑ 1 and µn → µ pointwise. Therefore, another appeal to Fatou’s

lemma implies (5.13), and hence the lemma.

Now we derive Theorem 5.2.

Proof of Theorem 5.2. Let us observe that by the Fubini–Tonelli theorem,

(5.36) Eκ∗ν(µ) = Eκ (µ , ν ∗ µ) .

In fact, both sides are equal to E[κ(X − X ′ − Y )], where (X ,X ′ , Y ) are independent, X

and X ′ are distributed as µ, and Y is distributed as ν.

Lemma 5.6, and polarization, together imply that the following holds for every Borel

probability measure σ on Rd:

(5.37) Eκ(µ , σ) =1

(2π)d

∫Rd

κ(ξ) σ(ξ) µ(ξ) dξ.

Indeed, we first notice that

(5.38) Eκ

(µ+ σ

2

)=

1

4Eκ(µ) +

1

4Eκ(σ) +

1

2Eκ(µ , σ).

Thus, we solve for Eκ(µ , σ) and apply Lemma 5.6 to deduce that

Eκ(µ , σ) =2

(2π)d

∫Rd

κ(ξ)

∣∣∣∣∣ (µ+ σ

2

)(ξ)

∣∣∣∣∣2

dξ

− 1

2(2π)d

∫Rd

κ(ξ) |µ(ξ)|2 dξ − 1

2(2π)d

∫Rd

κ(ξ) |σ(ξ)|2 dξ.

(5.39)

We solve to obtain (5.37).

Thus we define σ := ν ∗µ, which is a Borel probability measure on Rd. By (5.36), we have

Eκ∗ν(µ) = Eκ(µ , σ). We apply (5.37) to find that

Eκ∗ν(µ) =1

(2π)d

∫Rd

κ(ξ) σ(ξ) µ(ξ) dξ

=1

(2π)d

∫Rd

κ(ξ) ν(−ξ) |µ(ξ)|2 dξ.

(5.40)

In the last line, we have only used the fact that σ(ξ) = ν(−ξ)µ(ξ) for all ξ ∈ Rd. Because

the left-most term in (5.40) is real-valued, so is the right-most term. Therefore, we may

consider only the real part of the right-most item in (5.40), and this proves the result.

6. Absolute continuity considerations

Suppose E is a Borel measurable subset of RN that has positive Lebesgue measure, and

Y := Y (t)t∈E is an Rd-valued random field that is indexed by E.


Definition 6.1. We say that Y has a one-potential density v if v ∈ L1(Rd) is nonnegative

and satisfies the following for all Borel measurable functions f : Rd → R+:

(6.1)1∫

Ee−[s] ds

E

[∫E

f(Y (t))e−[t] dt

]=

∫Rd

f(x)v(x) dx.

In particular, X has a one-potential density v if for all Borel measurable functions f :

Rd → R+,

(Rf)(x) =

∫Rd

f(x+ y)v(y) dy

:= (f ∗ v)(x) for all x ∈ Rd,

(6.2)

where g(z) := g(−z) for all functions g. It follows from Lemma 2.1 in Hawkes [34] that X

has a one-potential density if and only if the operator R defined by (2.9) is strong Feller.

That is, if Rf is continuous whenever f is Borel measurable and has compact support.

In order to be concrete, we choose a “nice” version of the one-potential density v, when

it exists. Before we proceed further, let us observe that when v exists it is a probability

density. In particular, the Lebesgue density theorem tell us that

(6.3) v(x) = limε↓0

1

λd(B(0 , ε))

∫B(x,ε)

v(y) dy for almost all x ∈ Rd [λd].

We can recognize the integral as (R1B(0,ε))(−x). Moreover, we can alter v on a Lebesgue-null

set and still obtain a one-potential density for X. Therefore, from now on, we always choose

the following version of v:

(6.4) v(x) := lim infε↓0

(R1(0,ε)

)(−x)

λd(B(0 , ε))for all x ∈ Rd.

Lemma 2.3 states that the Fourier multiplier of R is KΨ. Therefore, the Fourier transform

of v is also KΨ; confer with (6.2). Because KΨ is real-valued, this proves the following:

Lemma 6.2. If X has a one-potential density v, then v = KΨ in the sense of Schwartz. In

particular, v is an integrable kernel of positive type on Rd.

Remark 6.3. Let ε1, . . . , εN denote N random variables, all independent of one another, as

well as XjNj=1 and X ′jNj=1, with Pε1 = ±1 = 1/2. We can re-organize the order of

integration a few times to find that

(6.5)1

2NE

[∫RN

f(X(t)

)e−[t] dt

]= E

[∫RN

+

f

(N∑j=1

εjXj(sj)

)e−[s] ds

].


Thus, in particular, X has a one-potential density if and only if the RN+ -indexed random field

(t1 , . . . , tN) 7→∑N

j=1 εjXj(tj) does [interpreted in the obvious sense]. Interestingly enough,

the latter random field appears earlier—though for quite different reasons as ours—in the

works of Marcus and Rosen [76,77].

Remark 6.4. It follows from the definitions that a random field Y (t)t∈E has a one-potential

density if and only if the Borel probability measure

(6.6) Rd ⊇ A 7→ E

[∫E

1A(Y (t))e−[t] dt

]is absolutely continuous.

Now suppose f ≥ 0 is Borel measurable. Then

(6.7)

∫RN

f(X(t)

)e−[t] dt ≥

∫RN

+

f(X(t))e−[t] dt.

We take expectations of both sides to deduce from (6.6) that if X has a one-potential density,

then so does X. When N = 1, it can be verified that the converse it also true. However, the

previous remark can be used to show that, when N ≥ 2, the converse is not necessarily true.

Thus, we can conclude that the existence of a one-potential density for X is a more stringent

condition than the existence of a one-potential density for X. Another consequence of the

preceding is the following: If the one-potential density of X exists and the potential density

of X is a.e.-positive, then the one-potential density of X is per force also a.e.-positive. We

have, and will, encounter these conditions several times.

Our next theorem is the main result of this subsection. From a technical point of view, it

is also a key result in this paper. In order to describe it properly, we introduce some notation

first.

If T is a nonempty subset of 1 , . . . , N, then we define |T | to be the cardinality of T , and

XT to be the subprocess associated to the index T . That is, XT is the following |T |-parameter,

Rd-valued random field:

(6.8) XT (t) :=∑j∈T

Xj(tj) for all t ∈ R|T |+ .

In order to obtain nice formulas, we define X∅ to be the constant 0. In this way, it follows

that, regardless of whether or not |T | > 0, each XT is itself an additive Levy process, and

the Levy exponent of XT is the function (Ψj)j∈T : Rd → C|T |. Thus, we can talk about the

stationary field XT for all T ⊆ 1 , . . . , N, etc. Despite this new notation, we continue to

write X and X in place of the more cumbersome X1,...,N and X1,...,N.

Our next result characterizes the positiveness of E[λd(X(RN

+ )⊕ F)]

in terms of the ca-

pacity of F defined in (5.6).


Theorem 6.5. Suppose there exists a nonrandom and nonempty subset T ⊆ 1 , . . . , N,such that XT has a one-potential density vT : Rd 7→ R+ that is continuous on Rd, and finite

on Rd \ 0. Then, X has a one-potential density v on Rd, and for all Borel sets F ⊆ Rd,

(6.9) E[λd(X(RN

+ )⊕ F)]> 0 ⇔ Cv(F ) > 0.

Proof. First, consider the case that T = 1 , . . . , N. In this case, X has a one-potential

density v := v1,...,N that is continuous on Rd, and finite away from the origin. In light

of Remark 5.3, Lemma 6.2 shows us that κ := v satisfies condition (2) of Theorem 5.2.

Theorem 5.2, in turn, implies that IΨ(µ) = Ev(µ) for all Borel probability measures µ on

Rd, and thence Cv(F ) = capΨ(F ). This and Theorem 1.1 together imply Theorem 6.5 in

the case that X has a continuous one-potential density that is finite on Rd \ 0.Next, consider the remaining case that 1 ≤ |T | ≤ N − 1. For all Borel sets A ⊆ Rd define

(6.10) VT c(A) :=1

2N−|T |E

[∫RN−|T |

1A

(XT c(s)

)e−[s] ds

].

This is the one-potential measure for the stationary field based on the additive Levy process

XT c . The corresponding object for XT can be defined likewise, viz.,

(6.11) VT (A) :=1

2|T |E

[∫R|T |

1A

(XT (t)

)e−[t] dt

].

We can observe that

VT c(A) := E

[∫RN

+

1A

(∑j∈T c

εjXj(sj)

)e−[s] ds

],

VT (A) := E

[∫RN

+

1A

(∑j∈T

εjXj(sj)

)e−[s] ds

],

(6.12)

where ε1, . . . , εN are i.i.d. random functions that are totally independent of X1, . . . , XN and

X1, . . . , XN , and Pε1 = ±1 = 1/2.

Similarly, we can write for all Borel measurable functions f : Rd → R+ and x ∈ Rd,

(6.13) (Rf)(x) = E

[∫RN

+

f

(x+

N∑j=1

εjXj(tj)

)e−[t] dt

].

Note that the integral is now over RN+ [and not RN ]. Moreover, we can write

(Rf)(x) =

∫∫f(x+ y + z)VT (dy)VT c(dz)

=

∫f(x+ y) (VT ∗ VT c) (dx),

(6.14)


The condition that XT has a one-potential density vT is equivalent to the statement that

VT (dy)/dy = vT (y), whence it follows that (Rf)(x) =∫f(x+ y)v(y) dy, where

(6.15) v(y) :=

∫vT (y − z)VT c(dz).

This proves the assertion that X has a one-potential density v. Note that v is of the form κ∗ν,

where κ := vT and ν := VT c . Thus, another appeal to Theorem 5.2 shows that condition (2)

there is satisfied [confer also with Remark 5.3], and thence it follows that Cv(F ) = capΨ(F ).

This and Theorem 1.1 together complete the proof.

6.1. A relation to an intersection problem. Define for all Borel sets F ⊆ Rd,

HF :=x ∈ Rd : P

X((0 ,∞)N

)∩ (x⊕ F ) 6= ∅

> 0

HF :=x ∈ Rd : P

X(RN6=)∩ (x⊕ F ) 6= ∅

> 0,

(6.16)

where RN6= := t ∈ RN : tj 6= 0 for every j = 1, . . . , N. Clearly, HF ⊆ HF .

The following improves on Proposition 6.2 of Khoshnevisan, Xiao, and Zhong [68].

Proposition 6.6. Consider the following statements:

(0) E[λd(X(RN+ )⊕ F )] > 0;

(1) capΨ(F ) > 0;

(2) Cv(F ) > 0;

(3) λd(HF ) > 0;

(4) HF 6= ∅;

(5) HF = Rd; and

(6) HF = Rd.

Then:

(a) It is always the case that (0)⇔ (1).

(b) If X has a one-potential density, then (1)⇔ (3)⇔ (4).

(c) If X has an a.e.-positive one-potential density, then (1)⇔ (3)⇔ (4)⇔ (6).

(d) If there exists a non-empty T ⊆ 1 , . . . , N such that the sub-process XT has a one-

potential density vT , and vT is continuous on Rd and finite on Rd \ 0. Then,

(1)⇔ (2)⇔ (3)⇔ (4).

(e) Suppose the conditions of (d) are met, and let v denote the one-potential density of

X. If v > 0 a.e., then (1)⇔ (2)⇔ (3)⇔ (4)⇔ (5).


Remark 6.7. Proposition 6.6 is deceptively subtle. For example, The preceding proposition

might fail to hold if HF were replaced by the related set

(6.17) H∗F :=x ∈ Rd : P

X(RN

+

)∩ (x⊕ F ) 6= ∅

> 0.

Indeed, because X(0) = 0, it follows that H∗F = −F .

Now we prove Proposition 6.6.

Proof of Proposition 6.6. Part (a) is merely a restatement of Theorem 1.1.

We first observe the following consequence of the Fubini–Tonelli theorem:

(6.18) E[λd(X((0 ,∞)N

) F

)]=

∫Rd

PX((0 ,∞)N

)∩ (x⊕ F ) 6= ∅

dx.

See (2.7) for a similar computation. If capΨ(F ) > 0, then the left-hand side of (6.18) is

positive by a simple variant of Theorem 1.1. This proves that (1) ⇒ (3). Conversely, if

λd(HF ) > 0, then the right-hand side of (6.18)—and hence the left-hand side—are positive.

Another appeal to Theorem 1.1 shows us that (1)⇔ (3).

Now we finish the proof of (b). Clearly, (3) implies (4). In order to prove that (4)⇒ (3),

we define

(6.19) Qs := (s1 ,∞)× · · · × (sN ,∞) for all s ∈ RN .

We note that for all s ∈ RN and x ∈ Rd,

P

X(Qs) ∩ (x⊕ F ) 6= ∅

=

∫Rd

PX((0 ,∞)N

)∩ ((x− y)⊕ F ) 6= ∅

P

X(s) ∈ dy.

(6.20)

This uses only the fact that X(t)− X(s)t∈Qs is independent of X(s), and has the same law

as X. In particular,∫RN

P

X(Qs) ∩ (x⊕ F ) 6= ∅e−[s] ds

= 2N∫

Rd

PX((0 ,∞)N

)∩ ((x− y)⊕ F ) 6= ∅

v(y) dy,

(6.21)

where v denotes the one-potential density of X. [It exists thanks to Remark 6.4.]

If (4) holds, then there exists s ∈ RN+ such that PX(Qs) ∩ (x ⊕ F ) 6= ∅ > 0. Conse-

quently, PX(Qs)∩(x⊕F ) 6= ∅ > 0. As we decrease s, coordinatewise, the preceding prob-

ability increases. Therefore, the left-most term in (6.21) is (strictly) positive, and therefore

so is the right-most term in (6.21). We can conclude that PX((0 ,∞)N)∩(w⊕F ) 6= ∅ > 0

for all w in a non-null set [λd]. This proves (3)⇒ (4), whence (b).


In order to prove (c), let g denote the one-potential density of X. Then, we argue as we

did to prove (6.21), and deduce that∫RN

+

P X(Qs) ∩ (x⊕ F ) 6= ∅ e−[s] ds

=

∫Rd

PX((0 ,∞)N

)∩ ((x− y)⊕ F ) 6= ∅

g(y) dy.

(6.22)

If HF 6= Rd, then the left-most term must be zero for some x ∈ Rd. Because g > 0 a.e., this

proves that λd(HF ) = 0.

The equivalence of (1) ⇔ (2) in (d) is contained in Theorems 1.1 and 6.5. The rest of

Part (d) is proved similarly as in the proof of (b).

To prove (e): We note that “HF = Rd iff (4)” [under the condition that v > 0 a.e.] has a

very similar proof to “HF = Rd iff (4)” [under the condition that g > 0 a.e.], but uses (6.21)

instead of (6.22). We omit the details and conclude our proof.


We continue to let X1, . . . , XN denote N independent Levy processes on Rd with respective

exponents Ψ1, . . . ,ΨN . There is a large literature that is devoted to the “N -parameter Levy

process” ⊗Nj=1Xj defined by

(7.1)(⊗Nj=1Xj

)(t) :=

X1(t1)...

XN(tN)

for all t ∈ RN+ .

Note that:

(i) The state space of ⊗Nj=1Xj is (Rd)N ; and

(ii) In the special case that N = 2, (7.1) reduces to (1.2).

In this section we describe how this theory, and much more, is contained within the theory

of additive Levy processes of this paper.

Let us begin by making the observation that product Levy processes are in fact degenerate

additive Levy processes. Indeed, for all t ∈ RN+ we can write

(7.2)(⊗Nj=1Xj

)(t) = A1X1(t1) + · · ·+ANXN(tN),


where each Xj(tj) is viewed as a column vector with dimension d, and each Aj is the Nd×dmatrix

(7.3) Aj =

0(j−1)d×d

Id×d

0(N−j)d×d

.

Here: (i) 0(j−1)d×d is a (j − 1)d× d matrix of zeros; and (ii) Id×d is the identity matrix with

d2 entries. We emphasize that the matrix Id×d appears after j− 1 square matrices of size d2

whose entries are all zeroes.

It is also easy to describe the law of the process ⊗Nj=1Xj via the following characteristic-

function relation:

(7.4) E

[exp

iN∑j=1

ξj ·Xj(tj)

]= exp

(−

N∑j=1

tjΨj(ξj)

),

valid for all ξ1, . . . , ξN ∈ Rd and t1, . . . , tN ≥ 0. Therefore, the exponent of the N -parameter

additive Levy process ⊗Nj=1Xj is

(7.5) Ψ(ξ1, . . . , ξN

)=

Ψ1(ξ1)...

ΨN(ξN)

for all ξ1, . . . , ξN ∈ Rd.

Therefore, Theorem 1.1 applies, without further thought, to yield the following.

Corollary 7.1. Choose and fix a Borel set F ⊆ (Rd)N . Then (⊗Nj=1Xj)(RN+ )⊕F has positive

Lebesgue measure with positive probability if and only if there exists a compact-support Borel

probability measure µ on F such that

(7.6)

∫(Rd)N

∣∣µ (ξ1, . . . , ξN)∣∣2 N∏

j=1

Re

(1

1 + Ψj(ξj)

)dξ <∞.

Next let us suppose that X1, . . . , XN have one-potential densities u1, . . . , uN , respectively.

According to Definition 6.1,

(7.7) E

[∫ ∞0

f(Xj(s))e−s ds

]=

∫Rd

f(a)uj(a) da,

for all j = 1, . . . , N , and all Borel measurable functions f : Rd → R+. [This agrees with the

usual nomenclature of probabilistic potential theory.]

Lemma 7.2. If X1, . . . , XN have respective one-potential densities u1, . . . , uN , then ⊗Nj=1Xj

has the one-potential density u(x1, . . . , xN) :=∏N

j=1 uj(xj) for all x1, . . . , xN ∈ Rd. Also,


⊗Nj=1Xj has the one-potential density

(7.8) v(x1, . . . , xN

):=

N∏j=1

(uj (xj) + uj (−xj)

2

)for all x1, . . . , xN ∈ Rd.

Finally, if uj(0) > 0 for all j, then u and v are strictly positive everywhere.

Proof. Let us prove the first assertion about the one-potential density of ⊗Nj=1Xj only. The

second assertion is proved very similarly.

We seek to establish that for all Borel measureable functions f : (Rd)N → R+,

(7.9) E

[∫RN

+

f((⊗Nj=1Xj

)(t))e−[t] dt

]=

∫(Rd)N

f(x1, . . . , xN

) N∏j=1

uj(xj)

dx.

A density argument reduces the problem to the case that f(x1, . . . , xN) has the special form∏Nj=1 fj(x

j), where fj : Rd → R+ is Borel measurable. But in this case, the claim follows

immediately from (7.7) and the independence of X1, . . . , XN .

In order to complete the proof, consider the case that uj(0) > 0. Lemma 3.2 of Evans [25]

asserts that uj(z) > 0 for all z ∈ Rd, whence follows the lemma.

The preceding lemma and Proposition 6.6 together imply, without any further effort, the

following two corollaries.

Corollary 7.3. Let X1, . . . , XN be independent Levy processes on Rd, and assume that each

Xj has a one-potential density uj such that uj(0) > 0. Then, for all Borel sets F ⊆ (Rd)N ,

(7.10) P(⊗Nj=1Xj

) ((0 ,∞)N

)∩ F 6= ∅

> 0


(7.11)

∫(Rd)N

∣∣µ (ξ1, . . . , ξN)∣∣2 N∏

j=1

Re

(1

1 + Ψj(ξj)

)dξ <∞.

Corollary 7.4 (Fitzsimmons and Salisbury, 1989). Suppose, in addition to the hypotheses

of Corollory 7.3, that each uj : Rd → R+ is continuous on Rd, and finite on Rd \0. Then,

for all Borel sets F ⊆ (Rd)N , (7.10) holds if and only if there exists a compact-support Borel

probability measure µ on F such that

(7.12)

∫∫ N∏j=1

(uj (xj − yj) + uj (yj − xj)

2

)µ(dx1 · · · dxN

)µ(dy1 · · · dyN

)<∞.

We can now prove Theorem 1.3 of the Introduction.


Proof of Theorem 1.3. We observe that:

(i) ∩Nj=1Xj((0 ,∞)) can intersect a set F ⊆ Rd if and only if (⊗Nj=1Xj)((0 ,∞)N) can

intersect G := x⊗ · · · ⊗ x : x ∈ F ⊆ (Rd)N ; and

(ii) any compact-support Borel probability measure σ on G manifestly has the form

σ(dx1 · · · dxN) = µ(dx1)∏N

j=2 δx1(dxj), where µ is a compact-support Borel prob-

ability measure on F , and hence σ(ξ1, . . . , ξN) is equal to µ(ξ1 + · · · + ξN) for all

ξ1, . . . , ξN ∈ Rd.

Therefore, Theorem 1.3 is a ready consequence of Corollaries 7.3 and 7.4.


First, we elaborate on the connection between Theorem 1.4 and Bertoin’s conjecture [4, p.

49] that was mentioned briefly in the Introduction. Recall that a real-valued Levy process

is a subordinator if its sample functions are monotone a.s.; see, for example, Bertoin [4, 6]

Fristedt [30], and Sato [93].

Remark 8.1. We consider the special case that S1 and S2 are two (increasing) subordinators

on R+ and F := 0, and define two independent Levy processes byX1 := S1 andX2 := −S2.

Evidently, X1(t1) + X2(t2) = 0 for some t1, t2 > 0 if and only if S1(t1) = S2(t2) for some

t1, t2 > 0.

Let Σj denote the one-potential measure of Sj, and suppose Σ1(dx)/dx = u1(x), where u1

is continuous on R and strictly positive on (0 ,∞). Let Uj denote the one-potential measure

of Xj. Then, U1 = Σ1 and U2 = Σ2, which we recall is the same as Σ2(−•). It follows then

that U1(dx)/dx = u1(x), and (u1 ∗ U2)(x) =∫∞

0u1(x + y) Σ2(dy) is strictly positive a.e.

Therefore, we may apply Theorem 1.4 with F := 0 to deduce that

(8.1) P S1(t1) = S2(t2) for some t1, t2 > 0 > 0 ⇔ Q(0) <∞.

Because U2 does not charge (−∞ , 0) in the present setting,

(8.2) Q(0) :=

∫ ∞0

[u1(y) + u1(−y)

2

]U2(dy) =

1

2

∫ ∞0

u1(y)U2(dy),

since u1(y) = 0 for all y < 0. [In fact, Lemma 3.2 of Evans [25] tells us that u1(y) = 0 for

all y ≤ 0.] Therefore, we conclude from (8.1) that

(8.3) P S1(t1) = S2(t2) for some t1, t2 > 0 = 0 ⇔∫ ∞

0

u1(y)U2(dy) =∞.

Define the zero-potential measures U0j as

(8.4) U0j (A) := E

[∫ ∞0

1A(Sj(t)) dt

],


for j = 1, 2 and Borel sets A ⊆ R+. Suppose U01 (dx)/dx = u0(x), where u0 is positive and

continuous on (0 ,∞). Then, it is possible to adapt our methods, without any difficulties, to

deduce also that

(8.5) P S1(t1) = S2(t2) for some t1, t2 > 0 = 0 ⇔∫ ∞

0

u01(y)U0

2 (dy) =∞.

The end of the proof of Theorem 1.5 contains a discussion which describes how similar

changes can be made to adapt the proofs from statements about one-potentials Uj to those

about zero-potentials U0j . We omit the details, as they are not enlightening.

We mention the adaptations to zero-potentials of (8.5) for historical interest: (8.5) was

conjectured by Bertoin, under precisely the stated conditions of this remark. Bertoin’s

conjecture [5] was motivated in part by the fact that, under the very same conditions as

above,

(8.6) P S1(t1) = S2(t2) for some t1, t2 > 0 = 0 ⇔ supz∈R

∫ ∞0

u01(y + z)U0

2 (dy) =∞.

It is possible to deduce (8.6), and the same statement without the zero superscripts, from

the present harmonic-analytic methods as well; see Lemma 5.4. The said extension goes

well beyond the theory of subordinators, and is a general sort of “low intensity maximum

principle”; see Salisbury [91]. But we will not describe the details further, since we find the

forms of (8.3) and (8.5) simpler to use, as well as easier to conceptualize.

Next we prove Theorem 1.4 without further ado.

Proof of Theorem 1.4. A direct computation reveals that the one-potential density of the

two-parameter additive Levy process X := X1 ⊕ X2 is u1 ∗ U2. Therefore, the equivalence

of (1.14) and (1.15) follows from Part (c) in Proposition 6.6. Next we note that the one-

potential density of X1 is described by

(8.7) v1(x) :=u1(x) + u1(−x)

2for all x ∈ Rd.

This function is positive a.e. and continuous away from the origin. We can use (6.15) and

verify directly that X has a one-potential density Q given by (1.17). Hence the last statement

follows from Part (e) of Proposition 6.6.

9. Intersections of Levy processes

The goal of this section is to prove Theorem 1.5. We first return briefly to Theorem 1.3

and discuss how it implies a necessary and sufficient condition for the existence of path-

intersections for N independent Levy processes. After proving Theorem 1.5, we conclude

this section by presenting a nontrivial, though simple, example.


9.1. Existence of intersections. First we develop some general results on equilibrium

measure that we believe might be of independent interest.

Choose and fix a compact set F ⊆ Rd, and recall that capΨ(F ) is the reciprocal of

inf IΨ(µ), where the infimum is taken over all Borel probability measures µ on F . It is not

hard to see that when capΨ(F ) > 0, this infimum is in fact achieved. Indeed, for all ε > 0

we can find a Borel probability measure µε on F such that

(9.1) IΨ(µε) ≤1 + ε

capΨ(F ).

We can extract any subsequential weak limit µ of µε’s. Evidently, µ is a Borel probability

measure on F , and

(9.2) IΨ(µ) ≤ 1

capΨ(F ),

by Fatou’s lemma. Because of the defining property of capacity, we also have

(9.3) IΨ(µ) ≥ 1

capΨ(F ).

Therefore, capΨ(F ) is in fact the reciprocal of the energy of µ.

Any Borel probability measure µ on F that has the preceding property is called an equilib-

rium measure on F . We now state and prove that there exists only one equilibrium measure

on a given compact set F .

Proposition 9.1. If F ⊆ Rd is compact and has positive capacity capΨ(F ), then there exists

a unique Borel probability measure eF on F such that

(9.4) capΨ(F ) =1

IΨ(eF ).

Proof. For all finite signed probability measures µ and ν on Rd define

(9.5) IΨ(µ , ν) :=1

(2π)d

∫Rd

(µ(ξ) ν(ξ) + ν(ξ) µ(ξ)

2

)KΨ(ξ) dξ.

This is well-defined, for example, if IΨ(|µ|) + IΨ(|ν|) <∞, where |µ| is the total variation of

µ. Indeed, we have the following Cauchy–Schwarz inequality: |IΨ(µ , ν)|2 ≤ IΨ(|µ|) · IΨ(|ν|).If σ is a finite [nonnegative] Borel measure on Rd, then IΨ(σ , σ) agrees with IΨ(σ), and this

is positive as long as σ is not the zero measure. However, we may note that slightly more

general fact that if σ is a non-zero finite signed measure, then we still have

(9.6) IΨ(σ , σ) > 0.

This follows from the fact that KΨ(ξ) > 0 for all ξ ∈ Rd.


Let c := 1/capΨ(F ), and suppose µ and ν were two distinct equilibrium measures on F .

That is, µ 6= ν but IΨ(µ) = IΨ(ν) = 1/c. In accord with (9.6),

(9.7) 0 < IΨ

(µ− ν

2,µ− ν

2

)=c− IΨ(µ , ν)

2.

Consequently, IΨ(µ , ν) < c, and hence

IΨ

(µ+ ν

2

)=

1

4IΨ(µ) +

1

4IΨ(ν) +

1

2IΨ(µ , ν)

=c+ IΨ(µ , ν)

2< c.

(9.8)

Because 12(µ + ν) is a Borel probability measure on F , this is contradicts the fact that c is

the smallest possible energy on F .

Next we note the following computation of the equilibrium measure in a specific class of

examples. The following result is related quite closely to the celebrated local ergodic theorem

of Csiszar [8]. [We hope to elaborate on this connection elsewhere.] See also Proposition A3

of Khoshnevisan, Xiao, and Zhong [68] for a related result.

Proposition 9.2. Suppose F is a fixed compact subset of Rd with a nonvoid interior. If

capΨ(F ) > 0, then eF is the normalized Lebesgue measure on F .

Proof. Our strategy is to prove that eF is translation invariant.

Let R(r) denote the collection of all closed “upright” cubes of the form

(9.9) I := [s1 , s1 + r]× · · · × [sd , sd + r] ⊆ F,

such that all of the si’s are rational numbers and r > 0. Each R(r) is a countable collection,

and hence we can (and will) enumerate its elements as I1(r), I2(r), . . . .

For all i, j ≥ 1 and r > 0 we choose and fix a one-to-one onto piecewise-linear map

θi,j,r : F → F that has the following properties:

• If a 6∈ Ii(r) ∪ Jj(r), then θi,j,r(a) = a;

• θi,j,r maps Ii(r) onto Ji(r) bijectively; and

• θi,j,r maps Ji(r) onto Ii(r) bijectively.

To be concrete, let us write Ii(r) = Ij(r) + b, where b ∈ Rd. Then we define

(9.10) θi,j,r(a) =

a− b if a ∈ Ii(r),

a+ b if a ∈ Ij(r),

a if a 6∈ Ii(r) ∪ Ji(r).

It can be verified that θi,j,r θi,j,r is the identity map.


For all integers n ≥ 1 and r > 0 consider

(9.11) ρn,r :=1

n2

∑∑1≤i,j≤n

(eF θ−1

i,j,r

).

Obviously, each ρn,r is a probability measure on F , and

(9.12) ρn,r(Ii(r)) = ρn,r(Ij(r)) for all 1 ≤ i, j ≤ n.

Furthermore, a direct computation reveals that for all i, j ≥ 1 and r > 0,

(9.13)∣∣∣ eF θ−1

i,j,r

∣∣∣ = |eF | pointwise.

Therefore, we can apply Minkowski’s inequality, to the norm µ 7→√IΨ(µ), to find that√

IΨ(ρn,r) ≤1

n2

∑∑1≤i,j≤n

√IΨ(eF θ−1

i,j,r

)=√IΨ(eF ).

(9.14)

By the uniqueness of equilibrium measure (Proposition 9.1), eF = ρn,r for all n ≥ 1 and

rationals r > 0. This and (9.12) together prove that eF (I) = eF (J) for all r > 0 and all

I, J ∈ R(r). A monotone-class argument reveals that for all Borel sets I ⊂ F and b ∈ Rd,

eF (I) = eF (b+I), provided that I, b+I ⊆ F . Because the Lebesgue measure is characterized

by its translation invariance, this implies the proposition.

Lemma 5.6 and Proposition 9.2, and Theorem 1.3 together imply the following variant of

a theorem of Fitzsimmons and Salisbury [29].

Corollary 9.3. Let X1, . . . , XN be independent Levy processes on Rd, and assume that each

Xj has a one-potential density uj that is continuous on Rd, positive at zero, and finite on

Rd \ 0. Then,

(9.15) P X1(t1) = · · · = XN(tN) for some t1, . . . , tN > 0 > 0

if and only if

(9.16)N∏j=1

(uj(•) + uj(−•)

2

)∈ L1

loc(Rd).

Proof. Clearly, (9.15) holds if and only if there exists n > 0 such that

(9.17) PX1(t1) = · · · = XN(tN) ∈ [−n , n]d for some t1, . . . , tN > 0

> 0.

Theorem 1.3 implies that (9.17) holds if and only if capΨ([−n , n]d) > 0. Lemma 5.6 and

Proposition 9.2 together prove the result.


9.2. Proof of Theorem 1.5. We begin by proving the first part; thus, we assume only that

the uj’s exist and are a.e.-positive.

Let S denote an independent M -parameter additive stable process of index α ∈ (0 , 2);

see (10.9). Next, we consider the (N +M)-parameter process Y := ⊗Nj=1(Xj −S); i.e.,

(9.18) Y(s⊗ t) :=

X1(s1)−S(t)...

XN(sN)−S(t)

for all s ∈ RN+ , t ∈ RM

+ .

It is not hard to adapt the discussion of the first few paragraphs in §7 to the present situation

and deduce that Y is an (N + M)-parameter additive Levy process, with values in (Rd)N ,

and that for all s ∈ RN+ , t ∈ RM

+ , and ξ := ξ1 ⊗ · · · ⊗ ξN ∈ (Rd)N ,

(9.19) E exp (iξ ·Y(s⊗ t)) = exp

(−

N∑k=1

skΨk(ξk)−

M∑l=1

tl∥∥ξ1 + · · ·+ ξN

∥∥α) .We can conclude readily from this that the characteristic exponent of Y is defined by

(9.20) Θ(ξ) :=

Ψ1(ξ1) , . . . ,ΨN(ξN) ,

∥∥∥∥∥N∑j=1

ξj

∥∥∥∥∥α

, . . . ,

∥∥∥∥∥N∑j=1

ξj

∥∥∥∥∥α

︸︷︷︸M times

,

for all ξ := ξ1 ⊗ · · · ⊗ ξN ∈ (Rd)N .

It follows readily from this and Lemma 7.2 that Y and Y both have positive one-potential

densities. Moreover, a direct computation involving the inversion formula reveals that the

potential density of Y is defined by

(9.21) v(x1 ⊗ · · · ⊗ xN) =

∫Rd

N∏j=1

(uj(x

j − y) + uj(y − xj)2

)w(y) dy,

for all x := (x1 ⊗ · · · ⊗ xN) ∈ (Rd)N . Here, w denotes the one-potential density of S. That

is,

(9.22) w(y) :=

∫RM

+

pt(y)e−[t] dt,

where pt denotes the density of S(t). That is, pt(y) := (2π)−d∫

Rd exp(−iy · z − [t]‖z‖α) dz

for all t ∈ RM+ and y ∈ Rd.

Since the uj’s are everywhere positive (Lemma 7.2), Proposition 6.6 tell us that 0 ∈Y(RN

+ ×RM+ ) with positive probability if and only if capΘ(0) > 0. Thus, the preceding


positive capacity condition is equivalent to the integrability of the function KΘ. That is,

0 ∈ Y(RN+ ×RM

+ ) with positive probability

⇔∫

(Rd)N

N∏j=1

Re

(1

1 + Ψj(ξj)

)dξ

1 + ‖ξ1 + · · ·+ ξN‖αM<∞.

(9.23)

On the other hand, it is manifestly the case that Y(RN+ × RM

+ ) contains the origin if and

only if the intersection of ∩Nj=1Xj(R+) and S(RM+ ) is nonempty. Thanks to Theorem 4.4.1

of Khoshnevisan [64, p. 428], for all Borel sets F ⊆ Rd,

(9.24) PS(RM+ ) ∩ F 6= ∅ > 0 if and only Cd−Mα(F ) > 0,

provided that we also assume that d > Mα. Suppose, then, that d > Mα. We can apply

the preceding, conditionally on X1, . . . , XN , and deduce that

0 ∈ Y(RN+ ×RM

+ ) with positive probability

⇔ Cd−Mα

(N⋂j=1

Xj(R+)

)> 0 with positive probability.

(9.25)

We compare the preceding display to (9.23), and choose M and α ∈ (0 , (d/M)∧2), such that

d−Mα is any prescribed number s ∈ (0 , d). This yields the following: For all s ∈ (0 , d),

Cs

(N⋂j=1

Xj(R+)

)> 0 with positive probability

⇔∫

(Rd)N

N∏j=1

Re

(1

1 + Ψj(ξj)

)dξ

1 + ‖ξ1 + · · ·+ ξN‖d−s<∞.

(9.26)

It follows from (9.26), Frostman’s theorem [64, p. 521] and an argument similar to the proof

of Theorem 3.2 in Khoshnevisan, Shieh and Xiao (2007) that the first identity of (1.18) holds

almost surely on ∩Nj=1Xj(R+) 6= ∅.In order to obtain (1.19), we assume also that the uj’s are continuous on Rd and finite on

Rd \ 0. Thanks to Proposition 6.6, Y(RN+ ×RM

+ ) contains zero with positive probability

if and only if v(0) <∞. Thus, (9.21) and (9.25) imply that

Cd−Mα

(N⋂j=1

Xj(R+)


⇔∫

Rd

N∏j=1

(uj(y) + uj(−y)

2

)w(y) dy <∞.

(9.27)


If we could replace w(y) by ‖y‖−d+αM , then we could finish the proof by choosing M and α

suitably, and then appealing to the Frostman theorem and the argument in Khoshnevisan,

Shieh and Xiao (2007). [This is how we completed the proof of the first part of the proof as

well.] Thus, our goal is to derive (9.27).

Unfortunately, we cannot simply replace w(y) by ‖y‖−d+αM by direct real-variable argu-

ments. Nonetheless, we recall the following fact from Khoshnevisan [64, Exercise 4.1.4, p.

423]: There exist c, C ∈ (0 ,∞) such that

(9.28) c‖y‖−d+Mα ≤ w(y) ≤ C‖y‖−d+Mα,

for all y ∈ (−1 , 1)d. Moreover, the upper bound holds for all y ∈ Rd (loc. cit., Eq. (2), p.

423); the lower bound [provably] does not. It follows then that∫Rd

N∏j=1

(uj(y) + uj(−y)

2

)dy

‖y‖d−Mα<∞

⇒ Cd−Mα

(N⋂j=1

Xj(R+)


⇒∫

(−1,1)d

N∏j=1

(uj(y) + uj(−y)

2

)dy

‖y‖d−Mα<∞.

(9.29)

If we could replace (−1 , 1)d by Rd in the last display, then our proof follows the outline

mentioned earlier. Thus, we merely point out how to derive (9.27), with w(y) replaced by

‖y‖−d+Mα, and omit the remainder of the argument.

The proof holds after we make a few modifications to the entire theory outlined here. We

describe them [very] briefly, since it is possible—though tedious—to check directly that the

present changes go through unhindered.

Define the operator R1⊗0 via

(9.30)(R1⊗0f

)(x) :=

1

2NE

[∫RN

e−[s] ds

∫RM

dt f(x+ Y(s⊗ t)

)]for all x ∈ (Rd)N .

If, we replace dt by exp(−[t])dt, then R1⊗0f gets turned in to Rf . As is, the operator R1⊗0f

fails to map Lp((Rd)N) into Lp((Rd)N). But this is a minor technical nuisance, since one

can check directly that

(9.31)(R1⊗0f

)(x) =

∫Rd

f(x+ y)v1⊗0(y) dy,

where v1⊗0 is defined exactly as v was, but with w(y) replaced by a certain constant times

‖y‖−d+Mα. And this shows fairly readily that if f is a compactly-supported function in


L1((Rd)N), then R1⊗0f ∈ L1loc((R

d)N). To complete our proof, we need to redevelop the

potential theory of the additive Levy process Y, but this time in terms of R1⊗0f and v1⊗0.

Let L1c((R

d)N) denote the collection of all compactly-supported elements of L1((Rd)N).

Because R1⊗0 maps L1c((R

d)N) to L1loc((R

d)N), the operator R1⊗0 has enough regularity that

we can adapt the Fourier analysis of the present paper without incurring any major changes.

And the end result is that Y(RN+ ×RM

+ ) contains zero if and only if v1⊗0(0) is finite. Now

we can complete the proof, but with v1⊗0 in place of v everywhere.

9.3. An example. Suppose X1, . . . , XN are independent isotropic stable processes in Rd

with respective Fourier transforms

(9.32) E exp(iξ ·Xj(t)) = exp(−cjt‖ξ‖αj),

for all t ≥ 0, ξ ∈ Rd, and 1 ≤ j ≤ N . Here, c1, . . . , cN are constants, and 0 < α1, . . . , αN <

2∧ d are the indices of stability. We are primarily interested in the case that N ≥ 2, but the

following remarks apply to the case N = 1 equally well.

In this section we work out some of the intersection properties of X1, . . . , XN . It is possible

to construct much more sophisticated examples. We study the present setting because it

provides us with the simplest nontrivial example of its type.

It is known that each Xj has a continuous positive one-potential density uj, and there

exist c, C ∈ (0 ,∞) such that

(9.33) c‖x‖−d+αj ≤ uj(x) ≤ C‖x‖−d+αj ,

for all x ∈ (−1 , 1)d and 1 ≤ j ≤ N . [These assertions follow, for example, from Corollary

3.2.1 on page 379, and Lemma 3.4.1 on page 383 of Khoshnevisan [64].] Consequently,

Theorem 1.3 immediately implies that for all Borel sets F ⊆ Rd,

(9.34) P

N⋂j=1

Xj(R+) ∩ F 6= ∅

> 0 ⇔ CNd−

PNj=1 αj

(F ) > 0.

In the case that N = 2, a slightly more general form of this was found in Khoshnevisan

[64, Theorem 4.4.1, p. 428] by using other methods. We may apply the preceding with

F = Rd, and appeal to Taylor’s theorem (loc. cit., Corollary 2.3.1, p. 525) to find that

P

N⋂k=1

Xk(R+) 6= ∅

> 0 ⇔ (N − 1)d <

N∑j=1

αj.(9.35)

Theorem 1.5, and a direct computation in polar coordinates, together show that the slack

in the preceding inequality determines the Hausdorff dimension of the set ∩Nk=1Xk(R+) of


intersection points. That is, almost surely on ∩Nk=1Xk(R+) 6= ∅,

(9.36) dimH

N⋂k=1

Xk(R+) =

[N∑k=1

αk − (N − 1)d

]+

.

This formula continues to hold in case some, or even all, of the αj’s are equal to or exceed

d. We omit the details.

9.4. Remarks on multiple points. Next we mention how the preceding fits in together

with the well-known conjecture of Hendricks and Taylor [41] that was solved in Fitzsimmons

and Salisbury [29], and also make a few related remarks.

Remark 9.4. Recall that a [single] Levy process X with values in Rd has N -multiple points if

and only if there exist times 0 < t1 < . . . < tN <∞ such that X(t1) = X(t2) = · · · = X(tN).

[We are ruling out the possibility that t1 = 0 merely to avoid degeneracies.]

By localization and the Markov property, X has N -multiple points almost surely if and

only if there are times t1, . . . , tN ∈ (0 ,∞) such that X1(t1) = · · · = XN(tN) with positive

probability, where X1, . . . , XN are N i.i.d. copies of X.

Suppose that X has a continuous and positive [equivalently, positive-at-zero] one-potential

density u that is finite on Rd\0. Then according to Corollary 9.3, X has N -multiple points

if and only if u(•) + u(−•) ∈ LNloc(Rd). An application of Holder’s inequality reveals then

that X has N -multiple points if and only if u ∈ LNloc(Rd). This is more or less the well-known

condition of Hendricks and Taylor [41].

More generally, the following can be deduced with no extra effort: Under the preceding

conditions, given a nonrandom Borel set F ⊆ Rd,

(9.37) P there exist 0 < t1 < · · · < tN such that X(t1) = · · · = X(tN) ∈ F > 0


(9.38)

∫∫ [u(x− y)

]Nµ(dx)µ(dy) <∞.

See, for example, Theorem 5.1 of Fitzsimmons and Salisbury [29].

Remark 9.5. We mention the following formula for the Hausdorff dimension of the N -multiple

points of a Levy process X: Under the conditions stated in Remark 9.4,

(9.39) dimHMN = sup

s ∈ (0 , d) :

∫Rd

[u(z)]N

‖z‖sdz <∞

a.s.,

where MN denotes the collection of all N -multiple points. That is, MN is the set of all

x ∈ Rd for which the cardinality of X−1(x) is at least N . This formula appears to be


new. Hawkes [36, Theorem 2] contains a similar formula—with∫

Rd replaced by∫

(−1,1)d—

which is shown to be valid for all isotropic [spherically symmetric] Levy processes that have

measurable transition densities.

In order to prove (9.39), we appeal to Theorem 1.5 and the Markov property to first

demonstrate that the dimension formula in (9.39) is valid almost surely on MN 6= ∅.Then the conclusion follows from the fact that the event MN 6= ∅ satisfies a zero-one law;

that zero-one law is itself proved by adapting the argument of Orey [80, p. 124] or Evans

[25, pp. 365–366]. We omit the details as the method is nowadays considered standard.

Remark 9.6. This is a natural place to complete a computation that we alluded to earlier

in Open Problem 4. Namely, we wish to prove that under the preceding conditions on the

Levy process X,

(9.40) dimHM2 = sup

s ∈ (0 , d) :

∫(−1,1)d

[u(z)]2

‖z‖sdz <∞

a.s.,

By the Markov property and an application of the Hewitt–Savage zero-one law, it suffices to

prove that almost surely on X1(R+) ∩X2(R+) 6= ∅,

(9.41) dimH

(X1(R+) ∩X2(R+)) = sup

s ∈ (0 , d) :

∫(−1,1)d

[u(z)]2

‖z‖sdz <∞

,

where X1 and X2 are independent copies of X.

We have seen already that if u 6∈ L2loc(R

d), then X1(R+)∩X2(R+) = ∅ almost surely, and

there is nothing left to prove. Thus, we may consider only the case that u ∈ L2loc(R

d). In this

case, we need to consider only the case that X1(R+)∩X2(R+) 6= ∅ with positive probability,

which as we have seen is equivalent to the condition that (X1X2)(R2+) contains the origin.

This and Theorem 1.1 together prove that KΨ ∈ L2(Rd), where KΨ(ξ) = Re(1 + Ψ(ξ))−1

and E exp(iξ ·X(t)) = exp(−tΨ(ξ)) in the present case. Now Lemma 6.2 shows that v—and

hence v—is square integrable, where v := 12(u(•) +u(−•)). That is, u ∈ L2(Rd). From here,

it is a simple matter to check that (9.39) with N = 2 implies (9.40).

10. Zero-one laws

We conclude this paper by deriving two zero-one laws: One for the Lebesgue measure

of the range of an additive Levy process; and another for the capacity of the range of an

additive Levy process.

The following was proved in Khoshnevisan, Xiao, and Zhong [68] under a mild technical

condition. Here we remove the technical condition (1.3) of that paper, and derive this result

as an elementary consequence of Theorem 1.1.


Proposition 10.1. Let X be a general N-parameter additive Levy process in Rd with expo-

nent Ψ. Then,

(10.1) E[λd(X(RN

+ ))]> 0 if and only if

∫Rd

N∏j=1

Re

(1

1 + Ψj(ξ)

)dξ <∞.

Suppose, in addition, that X has a one-potential density that is continuous away from the ori-

gin, and X has an a.e.-positive one-potential density. Then, with probability one, λd(X(RN+ ))

is zero or infinity.

Proof. According to Theorem 1.1, E[λd(X(RN+ ))] > 0 if and only if capΨ(0) > 0. Because

the only probability measure on 0 is δ0, it follows that capΨ(0) > 0 if and only if

KΨ ∈ L1(Rd). Thus follows (10.1).

The proof of Proposition 6.5 of Khoshnevisan, Xiao, and Zhong [68] goes through un-

hindered to conclude the remainder of our argument. We include it here for the sake of

completeness. Throughout the rest of this proof, we assume that X has an a.e.-positive

one-potential density.

The main step is to prove that if E[λd(X(RN+ ))] is finite, then it is zero. Note that for

every positive integer n,

E[λd(X(RN

+ ))]

≥ E[λd(X([0 , n]N)

)]+ E

[λd(X([n ,∞)N)

)]− E

[λd(X([0 , n]N) ∩ X′(RN

+ ))],

(10.2)

where X′ is an independent copy of X. Therefore, if E[λd(X(RN+ ))] is finite, then

(10.3) E[λd(X([0 , n]N

)]≤ E

[λd(X([0 , n]N) ∩ X′(RN

+ ))].

The inequality is, in fact, an equality. Let n ↑ ∞ to deduce that if E[λd(X(RN+ ))] is finite,

then E[λd(X(RN+ ))] = E[λd(X(RN

+ ) ∩ X′(RN+ ))]. Consider the function

(10.4) φ(a) := Pa ∈ X(RN

+ )

for all a ∈ Rd.

Then, we have just proved that∫

Rd φ(a) da =∫

Rd |φ(a)|2 da. Since 0 ≤ φ(a)(1 − φ(a)) ≤ 1

for all a ∈ Rd, it follows that φ ∈ 0 , 1 almost everywhere. Consequently, if E[λd(X(RN+ ))]

is finite, then

(10.5) E[λd(X(RN

+ ))]

= λd(φ−1(1)

).

According to Proposition 6.6, either φ(a) = 0 for all a, or φ(a) = 1 for all a. Since

E[λd(X(RN+ ))] < ∞, this and (10.5) together prove that φ(a) = 0 for all a, and hence


E[λd(X(RN+ ))] = 0. Finally the same argument as in the proof of Proposition 6.5 of Khosh-

nevisan, Xiao, and Zhong [68] shows that E[λd(X(RN+ ))] = ∞ implies λd(X(RN

+ )) = ∞a.s.

For all s > 0 define Cs(A) to be the s-dimensional Bessel–Riesz capacity of the Borel set

A ⊆ Rd. That is,

(10.6) Cs(A) :=

[inf

µ∈Pc(A)

∫∫µ(dx)µ(dy)

‖x− y‖s

]−1

,

where inf ∅ :=∞, 1/∞ := 0, and we recall that Pc(A) denotes the collection of all compact-

support Borel probability measures on A. Thus, Cs = Cκd−s , where κα(x) := ‖x‖−d+α denotes

the (d − α)-dimensional Riesz kernel [κα(0) := ∞]. The goal of this subsection is to derive

a zero-one law for the capacity of the range of an arbitrary additive Levy process X.

Proposition 10.2. If X denotes an N-parameter additive Levy process in Rd, then for all

β ∈ (0 , d) fixed, the chances are either zero or one that Cβ(X(RN+ )) is strictly positive.

The following formula for the Hausdorff dimension of X(RN+ ) is an immediate application

of Proposition 10.2 and the methods of Khoshnevisan and Xiao [66, Theorem 4.1]: If X is

an additive Levy process with values in Rd with characteristic exponent (Ψ1 , . . . ,ΨN), then

almost surely,

(10.7) dimH

(X(RN

+ ))

= sup

β ∈ (0 , d) :

∫Rd

N∏j=1

Re

(1

1 + Ψj(ξ)

)dξ

‖ξ‖d−β<∞

.

Here sup ∅ := 0.

Proof of Proposition 10.2. We choose and fix a β ∈ (0 , d), and assume that

(10.8) PCβ(X(RN+ )) > 0 > 0,

for there is nothing to prove otherwise.

Let us choose an integer M ≥ 1 and a real number α ∈ (0 , 2] such that β = d −Mα.

After enlarging the probability space, if need be, we may introduce M i.i.d. isotropic stable

processes S1, . . . , SM—independent also of all Xj’s—such that each Sj has stability index α.

In this way, we can consider also the M -parameter additive Levy process

(10.9) S(u) := S1(u1) + · · ·+ SM(uM) for all u := (u1 , . . . , uM) ∈ RM+ .

defined on Rd.

According to Theorem 7.2 of Khoshnevisan, Xiao, and Zhong [68],

PCβ

(X(RN

+ ))> 0> 0 ⇔ E

[λd(X(RN

+ )⊕S(RM+ ))]> 0.(10.10)


Because X⊕S is itself an (N+M)-parameter additive Levy process, Proposition 10.1 implies

that the right-hand side of (10.10) is equivalent to

(10.11)

∫Rd

(1

1 + ‖ξ‖α

)M N∏j=1

Re

(1

1 + Ψj(ξ)

)dξ <∞.

It remains to prove PCβ(X(RN+ )) > 0 = 1.

The proof is similar to that of Proposition 2.8 in Khoshnevisan, Xiao, and Zhong [67].

Define the random probability measure m on X(RN+ ) by

(10.12)

∫Rd

f(x)m(dx) =

∫RN

+

f(X(t)) e−[t] dt,

where f : Rd → R+ denotes a Borel measurable function. It follows from Lemma 5.6 and

the Fubini–Tonelli theorem that

(10.13) E

[∫∫m(dx)m(dy)

‖x− y‖β

]=

1

(2π)d

∫Rd

E(‖m(ξ)‖2) dξ

‖ξ‖d−β.

We may observe that

E(‖m(ξ)‖2) =

∫RN

+

∫RN

+

E[e−iξ·(X(t)−X(s))

]e−[s]−[t] ds dt

=N∏j=1

[∫ ∞0

∫ ∞0

e−sj−tj−|sj−tj |Ψj(sgn(sj−tj)ξ) dsj dtj

].

(10.14)

A direct computation shows that the preceding is equal to KΨ(ξ); see also the proof of

Lemma 3.4. Thanks to this and (10.11), the final integral in (10.13) is finite, whence it

follows that Cβ(X(RN+ )) > 0 almost surely.

Acknowledgements. A few years ago, Professor Jean Bertoin suggested to us a problem

that is addressed by Theorem 1.4 of the present paper. We thank him wholeheartedly.

We have been writing several versions of this paper since before April 12, 2005. During

the writing of one of these drafts we received several preprints by Dr. Ming Yang who,

among other things, has independently discovered Propositions 10.1 and 10.2 and equation

(10.7) of the present paper [103–105]. His method combines our earlier ideas [68] with an

elegant symmetrization idea, and is worthy of further investigation. We thank Dr. Yang for

communicating his work with us.

We are deeply indebted to an anonymous referee who read this paper surprisingly carefully

and in great detail. The present manuscript has benefitted a good deal from that referee’s

many corrections, suggestions, and generous remarks.


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Davar Khoshnevisan: Department of Mathematics, The University of Utah, 155 S. 1400 E.

Salt Lake City, UT 84112–0090

E-mail address: [email protected]: http://www.math.utah.edu/~davar


Department of Statistics and Probability, A-413 Wells Hall, Michigan State University,

East Lansing, MI 48824

E-mail address: [email protected]: http://www.stt.msu.edu/~xiaoyimi

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